A right triangle is a triangle in which one of the three angles is 90°. The triangle shown below is right-angled because the angle ⦣ACB between sides a and b is 90°. The side opposite the right angle, c, is called the hypotenuse.

One of the most important and best-known theorems of Euclidean geometry, the Pythagorean theorem, expresses a fundamental property of right-angled triangles. It states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides. The equation c² = a²+b² expresses this relationship in mathematical symbols.

Throughout history, countless proofs of the Pythagorean theorem have been given. Below we will examine some of the best known.

The simplest and most convincing proof

The simplest and most immediately convincing proof is a geometric one based on rearranging the position of four triangles within a square, as shown in the following image.

In the square on the left, four congruent right-angled triangles leave a central space of area c², corresponding to the square with the hypotenuse of each of the four triangles for its sides. The square on the right is the same as the other. The only difference is that the inner triangles have been moved so as to leave space for the formation of two internal squares, one of area a², which has sides equal to side a of each triangle, and the other of area b², whose sides are equal to side b of each triangle. The rearrangement of the triangles visually shows that the square c² is equal to the sum of the squares a² and b², as it was meant to prove.

Bhāskara’s proof

The same identity can also be proved algebraically. Let’s consider the square on the left in the image above. The length of each side is equal to the sum of sides a and b of the triangles inside it. Therefore, its area is equal to (a+b)². But we can also get the same area by adding the figures inside it, that is, the square of area c² and the four right-angled triangles. From this, it follows that the area c² of the inner square is equal to the area of the outer square (a+b)² minus the area of the four right triangles. Since the four triangles are congruent, we can write the equation:

Carrying out the calculations, we get:

Adding the like terms, it remains:

which is what we wanted to prove. This proof was given by the Indian mathematician and astronomer Bhāskara around 1150 CE.

Four triangles and a square

We can obtain another proof of the Pythagorean theorem by arranging the same four congruent right-angled triangles to form a square as in the following image.

In this case, the four triangles form a square whose sides are their hypotenuses, i.e., their sides c. Thus, the area of this square is c². The space left free by the triangles in the center of the figure forms a small square, the sides of which have a length equal to the difference between side a and side b of each triangle. Therefore, the area of this inner square is (a−b)². So, we can write the following equation:

Carrying out the calculations, we get:

Again, adding the like terms we get:

which is what we wanted to prove.

Garfield’s proof

The following proof of the Pythagorean theorem was published in 1876 in the New-England Journal of Education by James Garfield, who would become the twentieth president of the United States in 1881.

Let’s consider the figure below. If we take into account only the outer edge, it is a trapezoid (or trapezium), the area of which is calculated by multiplying the half sum of the lengths of the parallel sides by the height. In this case, the sum of the parallel sides is a+b. The height is also given by a+b. Therefore, the formula for the area of this trapezoid is:

On the other hand, if we consider its internal subdivision, the area of the figure is given by the sum of the areas of three right-angled triangles, two of which are congruent. So, the total area is equal to:

Thus, equating the area of the trapezoid to that of the three triangles, we can write the equation:

Finally, carrying out the calculations, we get:

which, once again, proves the identity established by the Pythagorean theorem.

Einstein’s proof

Based on the properties of similar triangles, one of the most elegant proofs of the Pythagorean theorem is due to Einstein, who discovered it when he was just twelve years old, according to the account given by the German physicist in his autobiography.

For this proof, the only figure needed is a right triangle, like the one in the image below.

If we draw the perpendicular h from vertex C to the opposite side c, the right-angled triangle ABC is divided into two triangles ADC and DBC, both right-angled, as shown in the following image. As can easily be deduced from the figure, the sum of the areas of the inner triangles ADC and DBC is equal to the area of the outer triangle ABC.

The key point in drawing the perpendicular h to side c of the triangle ABC is that the two internal triangles consequently obtained are similar to ABC. The hypotenuses of these two right triangles are the legs (side a and side b) of ABC. All this can be seen from the following image, in which the three triangles are drawn with the same spatial orientation and vertically aligned.

Similar triangles not only have their sides in proportion, but their areas are also in proportion, as well as the squares of each side. In particular, the ratio of the area of the triangle ABC to the square c² of its hypotenuse c is equal to the ratio of the area of the triangle ACD to the square a² of its hypotenuse a and the ratio of the area of the triangle CBD to the square b² of its hypotenuse b. If we call respectively X, Y, and Z the areas of the three triangles, the following equivalences hold:

Since Y+Z = X (i.e., the sum of the areas of the two inner triangles is equal to the area of the outer triangle), it follows that a²+b² = c².

We can algebraically derive this identity from the equation that relates the sum of the areas of the two inner triangles to the area of the outer triangle:

Since the three triangles are similar, we can replace in the equation the variables that designate the sides of the two inner triangles (d, h and e, h, respectively) with the variables a and b, which designate the sides of the outer triangle, multiplied by the constant of proportionality that expresses the ratio of side a to side c, and the ratio of side b to side c of the outer triangle:

Now, dividing both sides by ½ ab, we get:

Finally, multiplying both sides by c² to eliminate the denominators on the left, we get:

which is what we wanted to prove.

Euclid’s proof

This proof is given in Proposition 47 of Book 1 of Euclid’s Elements (about 300 CE).

Let ABC be a triangle with a right angle in vertex A. Let BDEC be the square, one of whose sides is the hypotenuse BC of the triangle, and let ACKH and BAGF be the squares one of whose sides are, respectively, the legs AC and BA of the right triangle. Now, let us draw from vertex A the perpendicular to the hypotenuse BC, which meets BC at point M, extending the perpendicular until it meets side DE of the square BDEC at point L. Finally, let us construct with suitable line segments the triangles FBC, ABD, AEC, and BCK, as shown in the following figure.

The final purpose of the proof is, as always, to show that the square that has the hypotenuse of the right triangle for its side (BDEC) is equal to the sum of the squares that have for their sides the two legs of the triangle (ACKH and BAGF). The way chosen by Euclid to prove this is to demonstrate that the two rectangles in which the square BDEC has been divided by the perpendicular conducted from the vertex A of the triangle, that is, LECM and DLMB, have area equal to the squares ACKH and BAGF respectively. Proving this also proves the Pythagorean theorem since the area of the square BDEC is equal to the sum of the areas of the two rectangles LECM and DLMB.

First, we note that the angle ⦣FBC is equal to the angle ⦣ABD. In fact, both angles are obtained by adding the same angle ⦣ABC to the right angles ⦣FBA and ⦣DBC. Also, sides FB and AB are the same because they are sides of the square BAGF. Similarly, sides BD and BC are the same because they belong to the square BDEC. Therefore, the triangles FBC and ABD are congruent since they have two equal sides and the angle between them is also equal.

We also note that FB is the base of the triangle FBC and FG (or BA) its height. Therefore, the area of ​​the triangle FBC is equal to ½(BF×FG). Since the area of ​​the square BAGF is equal to BF×FG, it follows that the area of ​​the triangle FBC is half that of the square BAGF.

Similarly, we note that BD is the base of the triangle ABD and BM (or DL) its height. Therefore, the area of ​​the triangle ABD is equal to ½(BD×BM). Since the area of ​​the rectangle BDLM is equal to BD×BM, it follows that the area of ​​the triangle ABD is half that of the rectangle BDLM.

But, as demonstrated above, the two triangles FBC and ABD are congruent. It follows that the area of ​​the square BAGF is equal to that of the rectangle DLMB.

With entirely analogous reasoning to that carried out so far, we can demonstrate that:

1. the triangles BCK and AEC are congruent because they have two sides and the angle between them equal;
2. the area of ​​the triangle BCK is half of the area of ​​the square ACKH;
3. the area of ​​the triangle AEC is half of the area of ​​the rectangle LECM;
4. the area of ​​the rectangle LECM is equal to that of the square ACKH.

Therefore, the area of ​​the square BDEC, which is the sum of the areas of the rectangles BDLM and LECM, is equal to the sum of the areas of the squares BAGF and ACKH, which is what we wanted to prove.

Proving the Pythagorean Theorem was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.

In the last fifty years, the ever more widespread diffusion of increasingly cheaper and more powerful home computers has made it possible to fully delegate to machines the execution of calculations that only humans could perform until recently. Among these calculations, we can certainly include finding the square roots of whole or decimal numbers. Few people today feel the need to learn how to compute a square root by hand, preferring to entrust the tedious task to software or a pocket calculator, digital instruments that are now within reach of practically anyone.

Be that as it may, here is the method for calculating the square root of a positive number to any desired level of accuracy for all those interested in learning how to do it. (There are also simpler methods based on estimates by trial-and-error, but we will leave the discussion for a future article.)

The calculation of a square root step by step

For this example exercise, we want to calculate the square root of 66,615.61:

First, let’s write this number by separating its digits into pairs so that the decimal point doesn’t occur in the middle of a pair. Only the first digit can be left unpaired if the number we are dividing into pairs has an odd number of digits before the decimal point, as in this case.

When calculating a square root, we proceed similarly to long division, but we always bring down two digits at a time, except if we have an unpaired first digit. In our example, the first goal is to find the highest positive integer, which, when squared, is less than or equal to 6. This number is 2 because 2 squared equals 4, while 3, the successor of 2, when squared is 9:

Let us draw a separating line above the radicand (the number we want to find the square root of), so we can write 2 above that line. 2 is the first digit of the number we are calculating, i.e., the square root of 66,615.61. Now, let us draw a vertical line to the left of the radicand, writing on its left 2×2, that is, the operation that allowed us to find 4 as the largest square less than or equal to 6. Then, we write 4 under the first digit of the radicand and perform the subtraction 6 – 4, writing the result under the number 4. Finally, we bring down the second pair of digits, 66, writing it to the right of 2, the result of the subtraction just performed. So, we get the number 266, as shown in the following image.

The next step is to multiply by 2 the first digit of our solution, 2, and write the result, 4, under the previous multiplication. In doing so, we leave two blanks next to 4 for a multiplication between two digits, as shown below.

Those two blanks must both be filled with the same number. Our goal is to find the multiplication, including the number 4 we wrote at the beginning, that yields the greater product equal to or less than 266. The number 6 is too big because 46×6 = 276. Instead, 5 is for us because 45×5 = 225. Therefore, we fill in the two empty spaces with 5 and write the multiplication result, 225, under 266. As in the previous case, now we perform a subtraction: 266 − 225 = 41. We write the difference under 225. The number 5, used for multiplying 45×5, is the second digit of the square root of 66,615.61. Thus, we write 5 to the right of 2 above the horizontal separation line.

We proceed by repeating the previous steps. We bring down the next pair of numbers in the radicand, 15, by writing it to the right of 41, the result of the last subtraction. We get the number 4,115. Now, we multiply 25 (the partial result of the calculation of the square root of 66,615.61) by 2 and write the product, 50, under the previous multiplication (45×5), leaving, as usual, two empty spaces for a number we will use for the next multiplication.

We have to find the largest number that is equal to or less than 4,115. The number that is right for us is 8. In fact, 508×8 = 4,064, while 509×9 = 4,581, a number greater than 4,115. Therefore, we write 8 in the two empty spaces and insert the multiplication result, 4,064, under 4,115. Then, we write 8 to the right of 5 above the radicand. Our partial result is 258. Now, it is important not to forget to enter the decimal point, which must be written to the right of the number 8, in correspondence with where the decimal point is in the radicand.

Finally, we perform the subtraction 4,115 − 4,064, getting 51 as the difference. So, we write 51 in the appropriate positions reserved for tens and units, as shown in the image below.

Now, we bring down the last pair of digits of the radicand, 61, by writing it next to 51. We get the number 5,161. Then, we multiply the partial result of our calculations, 258, by 2 and write the result, 516, to the left of the vertical line, in a position corresponding to 5,161. As always, we leave two empty spaces for the number that we will have to use for the next multiplication. This number must necessarily be 1 because it is the only number that allows us to obtain a product less than or equal to 5,161. In fact, 5,161×1 = 5,161. We write this product under the number 5,161 obtained by bringing down the last couple of digits of the radicand. Finally, we perform the subtraction 5,161–5,161, the result of which is 0.

Therefore, we write 1 as the last digit of our solution to the right of the decimal point. Since the remainder of the last subtraction is 0, it means our work is done. The square root of 66,615.61 is exactly 258.1.

A never-ending task

It should be noted that ending the calculation with a remainder of zero and an exact square root is an exception rather than the rule. Much more often, it happens that no subtraction yields a difference of zero. Therefore the calculation of the square root continues indefinitely, with the addition of new decimals. It is wise in these cases to interrupt the procedure when the square root obtained has a level of precision sufficient for any useful purpose.

For example, I report below the calculations performed to find the square root of 53,728, interrupted after reaching the fifth decimal place. Squaring the partial result of 231.79301, we get 53,727.99948…, only 52 hundred thousandths less than 53,728.

Two points deserve to be highlighted in the example above. After bringing down the last two digits of the radicand, i.e., 28, the square root calculation did not finish because we got a difference other than zero (367). In this case, you can continue the square root calculation, adding two zeros to the last difference, as shown in the above image. All solution digits calculated from this point on will be decimal.

The second point concerns the case in which the minuend is less than the subtrahend in the subtraction to be carried out. In the above example, this happened when the minuend was 515,100. In this case, the digit to add in the multiplication that yields the subtrahend must be zero. In this way, the product is zero, the subtrahend is zero, too, so the operation can continue. Zero will also be the digit to be added to the square root we are calculating.

Understanding the Mechanism

Why is the square root of a number computed exactly in the way we have illustrated so far? The reason can be understood by resorting to a geometric explanation. We can consider the radicand as the area of a square. If we call S² this area, then S is the measure of one side of the square and, also, the square root we are interested in finding.

If we divide each side of the square into two parts, one of length 10A and one of length B, the square’s area S² is equal to the square of the binomial:

The expansion of the square of this binomial is:

which can also be written as:

Since B is a common factor of the last two terms, we can rewrite the expression as:

The image below visually illustrates these geometric relationships.

Suppose we want to calculate the square root of 1024. First, we divide the radicand into two pairs of digits: 10 and 24. Then, we start the calculation by looking for the largest number which, when squared, is less than or equal to 10. Such number is 3, since 3² = 9 and 9 ≤ 10. Therefore, 3 is the first digit of the square root of 1024. At this point, we perform the subtraction 10−9 = 1 and bring down the other two digits of the radicand. So, we get the number 124.

Now let’s ask ourselves what relationship there is between 1024 and 124. It is easy to calculate that the difference of the two numbers is 900, which corresponds to 30². We note that 30 = 10×3. It corresponds to the segment of length 10A (with A=3), i.e., one of the two parts into which we divided the sides of the square of area S².

900 is also equal to 100×3², which corresponds to 100A², the area of the largest inner square in the figure above. In other words, subtracting 9 from 10 and then bringing the second pair of digits down is equivalent to subtracting 900 from 1024. It means that we calculated the area of the largest inner square (100A² = 900) and then subtracted it from the overall area of the square S² = 1024. Thus, 124 is the area of what remains of the square S² after subtracting the inner square 100A². This area consists of 2 rectangles of area 10AB and a square of area B². These relationships are illustrated in the following image.

At this point, we continue the calculation of the square root of 1024. We multiply by 2 the first digit of the solution, 3, getting 6. Then we add to the right of 6 two blanks separated by the multiplication sign. The blanks need to be filled by the same 1-digit number, with the ultimate goal of finding, through this multiplication, the largest number that is equal to or less than 124. The number we are looking for is 2 because 62×2 = 124.

Let us now ask ourselves what relationship the multiplication 62×2 has with the parts into which we have divided the square of area S²=1024. We know from the previous calculations that A=3 and the segment 10A=30. If we set B=2, then 62×2 corresponds exactly to [2(10A)+ B]B, i.e., the difference between ​​the square S² and the inner square 100A², shown in the above image.

In summary, the first operation to calculate a square root is to find the area of ​​the inner square 100A². The second operation consists of finding ​​the difference between the square of area S² (i.e., the radicand, the number we want to find the square root of) and the inner square of area 100A². This difference equals the sum of two rectangles of area 10AB and a square of area B². The common factor B allows us to transform this sum into the multiplication [2(10A)+B]B.

In this example, the remainder of 124–124 is zero. So the calculation is finished, and we are left with an exact square root: 32. In fact, 32²=1024.

Suppose, instead, that we want to calculate the square root of 1084. The calculations to be done are the same as the previous example up to the multiplication 62×2. This time, however, the subtraction to be made has not a zero remainder, since 184–124=60. This means that the square root of 1084 is a number greater than 32 but less than 33 (otherwise, the minuend would not have been 184 but 189, equal to 63×3).

Bringing all this back to our geometric explanation, having obtained a remainder of 60 indicates that with the last multiplication, 62×2, we could not calculate the total area of ​​the difference S²–100A². The partial result of the calculation of the square root of 1084, i.e., 32, corresponds to one side of a square of area 1024, equal to the sum of the inner square of area 100A² plus the two rectangles of area 10AB plus the square of area B². For now, an area X=60, equal to the sum of two rectangles of area (10A+B)C plus a square of area C², corresponding to the outermost of the two regions highlighted in the following image, remains out of the calculation.

To get a more precise solution, we add two zeros to 60, getting 6,000. At this point, we double the partial result, 32, and write the product, 64, on the left of the vertical line (see image below) leaving the usual two blanks for a 1-digit number. This number, added to 64 in the place of units, must be used for a multiplication with which we want to get the greater number that is equal to or less than 6,000. The 1-digit number we are searching for is 9. In fact, 649×9=5,841 and 5,841 ≤ 6,000. Thus, we have improved the calculation precision of the square root of 1084, bringing it from 32 to 32.9.

With the last multiplication, we got even closer to the total area of the square S². In fact, 32.9²=1082.41, just a bit less than 1084. This means that we have almost completely covered the outermost region highlighted in the above image, containing the two rectangles of area (10A+B)C plus the square of area C².

But, whenever there is a remainder, an area formed by the sum of two smaller and smaller rectangles and a square has remained out of the calculation. The mechanism for forming a new multiplication to increase the square root’s calculation precision is incremental. Multiplication 62×2 is equivalent to searching for the area of the section of the square S² corresponding to

The subsequent multiplication 649×9 tries to cover the area represented by the expression

If we continued the calculations in the search for even greater precision, the new multiplication should find the area represented by the expression

corresponding to two rectangles of area (10A+B+C)D plus a very small square of area D².

How to Calculate Square Roots by Hand was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.

A fateful letter

In 1903, the German logician and mathematician Gottlob Frege published, ten years after the first, the second volume of a work entitled Grundgesetze der Arithmetik (“Basic Laws of Arithmetic”), whose aim was to derive the laws of arithmetic from some logical axioms.

In contrast to this ambitious goal, or perhaps precisely because of that, we can read in the Appendix of that second volume a statement by Frege in some ways astonishing. With rare honesty, the author admits that the work he has just published, the fruit of years of hard study, contains a tremendous flaw that he does not quite know how to eliminate:

Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr Bertrand Russell, just when the printing of this volume was nearing its completion. It is a matter of my Axiom (V). I have never disguised from myself its lack of the self-evidence that belongs to the other axioms and that must properly be demanded of a logical law. (…) I should gladly have dispensed with this foundation if I had known of any substitute for it.

Bertrand Russell’s letter to Frege is dated June 16, 1902. The part of the letter that threw Frege into despair and confusion deserves to be quoted:

I find myself in full accord with you on all main points, especially in your rejection of any psychological element in logic (…). On functions in particular I have been led independently to the same views even in detail. I have encountered a difficulty only on one point. You assert (p. 17) that a function could also constitute the indefinite element. This is what I used to believe, but this view now seems to me dubious because of the following contradiction: Let w be the predicate of being a predicate which cannot be predicated of itself. Can w be predicated of itself? From either answer follows its contradictory. We must therefore conclude that w is not a predicate. Likewise, there is no class (as a whole) of those classes which, as wholes, are not members of themselves. From this I conclude that under certain circumstances a definable set does not form a whole.

Cantor’s definition of set and Russell’s paradox

The contradiction that Russell pointed out had to do with the very concept of what can be considered a set and what cannot. According to the definition of the creator of set theory, the German mathematician Georg Cantor,

By a “set” we mean any collection M into a whole of definite, distinct objects m (which are called the “elements” of M) of our perception or of our thought.

Cantor’s definition left room for the possibility of considering as a set any collection of objects of perception or thought based on any conceivable property. According to such definition, even extremely abstract objects of thought, like, for example, the whole of all existing things, could rightly be considered a set.

So, if we call U the set of all existing things, both those endowed with physical reality and those simply thought of, it follows that U plays a double role. As a set, it is the container of this totality of objects. But it is also a content of itself because it is an existing thing, albeit as a pure object of thought. And, if a set can contain itself among its elements, then it is also possible to conceive the set of all sets that are elements of themselves. Let Y be such a set. In symbols it can be defined as:

But this overlapping of roles between content and container ended up turning into a lethal weapon for the logical coherence of a theory that allowed any property to be used to build a set. The contradiction Russell speaks of in his letter to Frege arises from a definition of a set which is opposite and yet extremely similar to the set defined above: The set of all sets that are NOT elements of themselves. Let W be such a set. In symbols, it corresponds to:

The contradiction follows directly from the very definition of such a set. If W is the set of all sets that are not elements of themselves, is W or not an element of itself? If W is NOT an element of itself, then it SHOULD BE an element of itself under its definition. But, if W IS an element of itself, then W no longer matches its definition. Both alternatives inevitably lead to a contradiction. In symbols:

The formula states that defining W as the set of all sets that are not elements of themselves implies that W is an element of itself if and only if W is NOT an element of itself, which is openly contradictory.

Russell’s view of the paradox

According to Russell, the contradiction he notified to Frege arose because the logic underlying the statements describing a class of objects that contains or does not contain itself was inadequate. It is interesting to read what Russell wrote about this problem in “The Philosophy of Logical Atomism” (1918):

(…) the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just a noise without meaning. That has to do with the fact that classes (…) are incomplete symbols (…) ; you are talking nonsense when you ask yourself whether a class is or is not a member of itself, because in any full statement of what is meant by a proposition which seems to be about a class, you will find that the class is not mentioned at all and that there is nothing about a class in that statement. It is absolutely necessary, if a statement about a class is to be significant and not pure nonsense, that it should be capable of being translated into a form in which it does not mention the class at all. This sort of statement, “Such-and-such a class is or is not a member of itself”, will not be capable of that kind of translation. (…) the symbol for a class is an incomplete symbol; it does not really stand for part of the propositions in which symbolically it occurs, but in the right analysis of those propositions that symbol has been broken up and disappeared.

Russell’s solution

Russell’s solution to the paradox that bears his name was the so-called theory of types. In this theory, classes are formed hierarchically, starting from the simplest, created to describe particular objects. More abstract classes only make sense if they are properly formed from lower-level classes. The following quote is also taken from “The Philosophy of Logical Atomism” (1918):

For instance, the totality of classes in the world cannot be a class in the same sense in which they are. So we shall have to distinguish a hierarchy of classes. We will start with the classes that are composed entirely of particulars: that will be the first type of classes. Then we will go on to classes whose members are classes of the first type: that will be the second type. Then we will go on to classes whose members are classes of the second type: that will be the third type, and so on. Never is it possible for a class of one type either to be or not to be identical with a class of another type.

In such a hierarchical construction, the class of all the classes which are not elements of themselves is pure logical nonsense; it cannot exist because it cannot be constructed correctly from a lower hierarchical level. Charles C. Pinter, a professor at the Bucknell University, briefly but clearly explains the basic mechanism of the theory of types in his “A Book of Set Theory” (2014):

(…) in his efforts to reinstate intuitive set theory, Russell was led to the idea that we may consider sets to be ordered in a hierarchy of “levels,” where, if A and B are sets and A is an element of B, then is B “one level higher” than A. (…) Every set has a natural number assigned to it, called its level. The simplest sets, those of level 0, are called individuals—they do not have elements. A collection of individuals is a set of level 1; a collection of sets of level 1 is a set of level 2; and so on. In the theory of types the expression a ∈ B is only meaningful if, for some number n, a is a set of level n and B is a set of level n+1. It follows that the statement x ∈ x has no meaning in the theory of types, and as a result, Russell’s paradox vanishes for the simple reason that it cannot even be formulated.

However, as Pinter explains in his book, the theory of types, due to a certain heaviness and the numerous assumptions to which Russell was forced, has never managed to gain widespread favor among mathematicians:

Russell’s theory of types is built upon a beautifully simple idea. Unfortunately, in order to make it “work,” Russell was forced to add a host of new assumptions, until finally the resulting theory became too cumbersome to work with and too complicated to be truly pleasing. (…) As a result of these shortcomings, the theory of types has not gained wide acceptance among mathematicians, although it is still an interesting (and maybe promising) area of research.

Zermelo-Fraenkel’s solution to Russell’s paradox

The solution to Russell’s paradox that has been most successful and widely accepted today is that deriving from the axiomatic set theory known as ZFC. In this acronym, the Z stands for Zermelo (the German logician and mathematician Ernst Zermelo, who provided the first axiomatization of the theory in 1908); the F stands for Fraenkel (the mathematician Abraham Fraenkel, author of important contributions to Zermelo’s theory); finally, the C stands for “choice,” referring to the axiom of choice, formulated in 1904 by Zermelo and today considered an integral part of ZFC axiomatic theory.

In the cited “A Book of Set Theory” (2014), Pinter explains synthetically how Zermelo’s theory works:

In Zermelo’s system, there is one primitive relation, denoted by the symbol ∈; the expression x ∈ Y is to be read “x is an element of Y ”. The variables x, y, z, X, Y, etc., which we place to the right or to the left of the symbol ∈, stand for objects which we agree to call “sets.”

(…) The reader may feel there ought to be two kinds of objects, namely sets and elements. Actually, this distinction is unnecessary (…); a useful simplification which is made in axiomatic set theory is to regard the elements of every set to be sets themselves; in other words, every set is considered to be a set of sets.

(…) Almost every set that arises in our thinking is a set consisting of all the objects of a specified kind — that is, consisting of all the objects which satisfy a given condition. This is the most natural way in which sets occur: we are able to describe a condition on x — let us symbolize this condition by S(x) — and we are led to speak of the set of all objects x which satisfy S(x).

(…) Since this is the most natural way that sets arise, it is clearly desirable to have a principle in set theory which makes it possible — given any condition S(x) — to form the set of all objects x which satisfy S(x). However, if such a principle is adopted without any restrictions, we are led to the paradoxes (for example, we can form the set of all sets which are not elements of themselves). Thus, we must devise such restrictions on this principle as will eliminate the paradoxes. Zermelo conceived the following restriction: Let S(x) be a condition on x; we cannot form the set of all x which satisfy S(x); but, if A is a given set, we can form the set of all x in A which satisfy S(x). Thus, roughly speaking, a property of objects cannot be used to form a “new” set, but only to “select,” from a set A whose existence has already been secured, all the elements which satisfy the given property.

According to its function, Zermelo called this principle the axiom of selection and stated it as follows:

Let A be a set, and let S(x) be a statement about x which is meaningful for every object x in A. There exists a set which consists of exactly those elements in x and A which satisfy S(x).

In other words, the axiom of selection allows associating a set with a given property only if this set is an element of another set, whose existence is already granted by the axioms defined in the ZFC system. In symbols,

to be read: “the set of all x in A such that S(x)”. The point worth noting is that Zermelo’s theory does NOT allow directly creating the set of all sets endowed with a certain property:

to be read: “the set of all x which satisfy S(x)”.

Finally, let us follow Pinter’s account of how the restriction introduced by Zermelo avoids Russell’s paradox:

First, let us see what happens to Russell’s paradox: the crucial set in Russell’s argument is the “set of all sets which are not elements of themselves,” which can be symbolized as {x | x ∉ x}. As we have noted, this set cannot be formed in Zermelo’s system; the best we can do is to produce {x ∈ A | x ∉ x}, where A is any set which can be shown to exist. If we substitute {x ∈ A |x ∉ x} for {x | x ∉ x} in Russell’s argument, the outcome changes completely. Indeed, let us go through the steps of the argument, with S denoting {x ∈ A | x ∉ x}:

S ∈ S is impossible, for S ∈ S implies S ∉ S, a contradiction! Thus S ∉ S. It follows that S ∉ A, for if S were in A, then (because S ∉ S) we would have S ∈ S, which would be a contradiction.

Hence Russell’s argument merely proves that if A is any set, then the set {x ∈ A | x ∉ x} cannot be an element of A.

The architect’s view of set theory

ZFC axiomatic theory resembles Euclidean geometry in some respects. Just as Euclidean theorems are deductively derived from a few undefined principles, similarly, the ZFC theory does not focus on what sets are philosophically but on how it is possible to construct ever larger and more complex sets in a non-contradictory way, starting from the simplest possible set, that is, the empty set. This way of proceeding, similar to that by which an architect designs and constructs a building, is sufficient to solve Russell’s paradox and, more importantly, to provide a coherent foundation for the whole construction of mathematics. Winfried Just and Martin Weese explain this clearly and briefly in the first volume of their “Discovering Modern Set Theory” (1996):

Think of the elements of a set as building blocks, and the formation of a set as assembling these building blocks into a whole. It is reasonable to require that at the moment a given set M is being formed, all its building blocks must have already attained their final shape; in this sense they should be “definite.” Let us call this stance the architect’s view of set theory. It stipulates that although it is possible to contemplate all sets at once, each set has to be formed at some moment in an abstract “time,” and at that moment, all its building blocks must already have been available in their final shape. Also, once a set is formed, one should be able to use it as a building block of other sets.

This view solves Russell’s Paradox in an unexpected way: M is not a set, because it could never have been assembled! At no moment in set-theoretic time do all the building blocks for the construction of M exist.

How can the architect’s view of set theory be expressed with sufficient mathematical precision? The approach concentrates not on what sets are, but on how sets are being formed. At the beginning of set-theoretic time, the only set that can be formed is the empty set, since no previous building blocks exist. Once this set is formed, it can be used as a building block for further sets. The modern alternative to Cantor’s definition is to describe precisely by which operations new sets can be built from existing ones, and then to apply these operations successively to the empty set.

Can we get all sets in this way? Perhaps not, but we can construct a universe of sets rich enough to encompass all known mathematics. This will do for starters. The architect’s view of set theory can be formalized by axioms, similar to the way in which our space intuitions were formalized by Euclid more than two thousand years ago. (…) Once an axiom system has been formulated, one can ask whether a given mathematical statement or its negation follows from the axioms. The answer may be a ‘’yes,’’ a “no,” or an independence result.

Russell’s Paradox was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.

Equivalence relations and equivalence classes

Given a nonempty set A, a relation R in A is a subset of the Cartesian product A×A. An equivalence relation, denoted usually with the symbol ~, is a relation in A, which is reflexive, symmetric, and transitive. More specifically, an equivalence relation ~ in A is:

• reflexive if x~x for every x ∈ A (any x element of A is equivalent to itself);
• symmetric if, for every x, y ∈ A, x~y implies y~x (any x equivalent to y in A, implies that y is equivalent to x);
• transitive if, for every x, y, z ∈ A, x~y and y~z imply x~z (if x is equivalent to y and y is equivalent to z, then x is equivalent to z).

If ~ is an equivalence relation on a set A, the equivalence class of an element x of A under the equivalence relation ~ is the set

This formula means that x and y, both elements of A, can belong to the same equivalence class if and only if they are equivalent. Stated in other words, an equivalence class can contain only elements sharing the same equivalence relation.

Partitioning a set through equivalence relations

A partition P of a set A is a set of subsets of A with the following three properties:

• it is disjoint, i.e., no two different sets in P have any common elements; therefore, if C and D are any two distinct subsets of P, the intersection of C and D is the empty set;
• it is exhaustive, i.e., each element of A is in some set in P; therefore, the union of all subsets of P is equal to A;
• No set in P is empty.

Any equivalence relation on a set A creates a partition P of A whose elements are nonempty subsets of A formed by the equivalence classes based on that relation.

For example, let A be the set of the standard 52 cards in a deck of French-suited cards, and let ~ be the equivalence relation “has the same rank as.” This equivalence relation produces thirteen distinct equivalence classes, each for any rank from aces to kings. In this way, from a single set A made up of 52 elements (all the cards in the deck), we arrive at a partition P of A, consisting of the thirteen subsets shown in the following image.

Each of the thirteen boxes represents a subset in P. As you can easily verify, P, i.e., the union of all thirteen subsets, is a partition of A, since:

• no subset in P is empty;
• no elements (the single cards of the deck) is found in more than one subset;
• all the elements of A are in P, so the union of all subsets of P is equal to A.

The above is in no way the only partition P of A we can create using an equivalence relation. Another trivial partition is that based on the equivalence relation ~ on A “has the same suit as.”

This subdivision of A is also a partition of A as the above, since it meets the three required criteria:

1. no empty set;
2. no elements in more than one set (all subsets of P are pairwise disjoint);
3. no elements of A left out (P is exhaustive).

A set consisting of all the subsets of a given set induced by an equivalence relation on that set is also called a quotient set. Given a set A and an equivalence relation ~ on A, the quotient set, denoted A\~, consists of all the equivalence classes based on that relation:

A powerful categorization tool

Equivalence relations, equivalence classes, and partitions are powerful tools for organizing sets of any kind into categories, based on the sharing of specific properties by the elements that belong to a set.

Once a property has been defined through an equivalence relation, all the elements of a set can be divided into disjoint and nonempty subsets, each characterized by having or not having that particular property. For example, a city’s population can be divided by age, gender, educational level, eye color, wealth, etc. Each of these properties can be defined by an equivalence relation, thanks to which the original set, i.e., the total population, can be divided into homogeneous groups. A partition is nothing more than a way of dividing the elements of a set into groups, that is, subsets, homogeneous concerning a particular property.

Any member of an equivalence class defined by a particular equivalence relation can represent that class. For example, in a deck of French-suited playing cards, any one of the thirteen cards of hearts is a fully-fledged representative of that equivalence class.

Equivalence Relations, Equivalence Classes and Partitions was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.

Trichotomy

Let us consider the case in which x and y are two distinct elements of a set A. Given a binary relation R on A, there are orderings of the set requiring that exactly one of the following three alternatives is true with respect to that relation:

(xRy means that x is related to y by the relation R; symmetrically, yRx means that y is related to x by the relation R.) Trichotomy is the technical term used to define the condition where exactly one of three alternatives must be true.

Total or linear ordering

Trichotomy plays a role in defining the properties of an ordering relation called total ordering. Given any distinct x and y in a set A, this relation tells us just which one is smaller. First of all, no element of A can be smaller than itself. Furthermore, if x is less than y and y is less than z, then x should be less than z. Those properties participate in the definition of total ordering.

Thus, a total (or linear) ordering on a set A is a binary relation R on A that obeys the following conditions:

1. R is antireflexive: There is no x in A for which xRx;
2. R is transitive: If xRy and yRz, then xRz;
3. R satisfies trichotomy. As explained above, this means that only one among xRy, x=y, and yRx is true. More specifically, for any distinct x and y in A, either xRy or yRx is true, but not both. In this case, R is said to be connected on A.

Total ordering is also called linear ordering because the binary relations of this kind on a given set cannot create circles. To clarify the point, let us consider the following relations:

Since total ordering is a transitive relation, then

But the last relation should infringe the first condition: No x for which xRx is true can be an element of A if A is a totally ordered set.

All that being said, R is certainly not the most used symbol to define this relation in the context of a comparison of numbers. The symbol normally used is instead <. For example,

are three total ordering relations on the set ℕ of natural numbers. The same relations can be symbolically expressed in non-abbreviated form as:

Therefore, < is a subset of ℕ and precisely the set that contains the above three ordered pairs:

The total ordering defined by these three relations can be represented graphically as a series of three equally spaced points on a line segment, a representation from which we can deduce why this ordering is also called linear.

The graph conveys the same information defined above symbolically: 3 is less than 4 and less than 5, 4 is less than 5.

Partial ordering

A relation R on a set A is a partial ordering if and only if:

• R is reflexive (for all x elements of A, xRx);
• R is antisymmetric (for all x, y elements of A, if xRy and yRx, then x=y);
• R is transitive (for all x, y and z elements of A, if xRy and yRz, then xRz).

The divisibility relation over positive integers is an example of partial ordering. The following diagram shows a very little part of what would be an infinite structure.

Another example of partial ordering is the inclusion ordering of all the subsets of a given set (a relation defined by the symbol ⊆). The following Hasse diagram shows the inclusion ordering on the set {0, 1, 2}.

This kind of diagram is a simplification in which the reflexive relations of each element with itself are not shown. Likewise, the transitive relations (the arcs linking any element a to c, both related to a common element b) are not shown. The diagram only shows the direct ordering relations from any element a to any element b, such that no intermediate element i exists between a and b. The lines indicating these relations point all upwards, so there is no need to use arrows suggesting the direction of each relation.

Even though with this simplification, Hasse diagrams perfectly capture the distinctive features of partial orderings. As can be easily noted observing the above figures, each element of a partially ordered set can be related to many elements of the same set. Moreover, two distinct elements can be totally unrelated. For instance, no line connects the positive integers 3 and 4 in the diagram showing the divisibility relation. Also, no line connects 6 and 7 in the same diagram. In fact, 4 is not divisible by 3. Similarly, 7 is not divisible by 6.

These two features differentiate partial ordering from total ordering. All the elements of a totally ordered set are connected in a series forming a linear chain. Instead, the elements of a partially ordered set form branches in which the same element can be directly connected to many other elements, maintaining the possibility to be unrelated to other elements of the same set.

Comparable and incomparable elements of a set

A binary relation R on a set A is a subset of the Cartesian product A×A. Given two elements x, y ∈ A, x is said to be comparable to y with respect to the binary relation R if either xRy or yRx is true. Conversely, x is said to be incomparable to y with respect to the same relation R if neither xRy nor yRx is true.

Let us consider, for example, the natural numbers 19 and 21 with respect to the ordering relation <. Since 19 < 21, then 19 and 21 are comparable with respect to the binary relation <.

Let us consider, instead, the elements {0} and {1} of the power set ℘({0,1,2}) with respect to the partial order ⊆. Since they are not subsets of each other, {0} and {1} are incomparable with respect to ⊆.

The distinction between comparable and incomparable elements of a set helps to clarify the difference between total ordering and partial ordering. A totally ordered set is a partially ordered set in which any two elements are comparable. From this definition, it follows, for example, that natural numbers are a totally ordered set with respect to the binary relations < (‘less than’) and > (‘greater than’) while they are a partially ordered set with respect to the relation | of divisibility.

Posets and tosets

The symbol ⪯ is usually used to denote a partial ordering. A partially ordered set or poset is a set together with a binary relation indicating that one of the elements precedes the other in the ordering for certain pairs of elements in the set. Therefore, if A is a set, a poset is:

Another example of a poset is the set of natural numbers together with the binary relation of divisibility:

Similarly, a totally ordered set, sometimes called a toset, is a set together with a binary relation indicating that one of the elements precedes the other in the ordering for any pair of elements in the set. Some authors use the symbol ⊑ to denote this relation. Thus, if A is a set, a toset is:

Natural numbers are an example of a totally ordered set with respect to the binary relation <:

Ordering Relations was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.

A fundamental feature of mathematics is the possibility of exploiting the link between algebra and geometry made visible by the representation of objects such as straight lines, curves, circles, etc., in the so-called Cartesian plane. This system is so effective because each point of the plane can be unambiguously described by a pair of coordinates. The x-coordinate defines the position of each point to the horizontal axis or abscissa; the y-coordinate defines this position to the vertical axis or ordinate.

The set-theoretic definition of ordered pair

The attempt undertaken at the beginning of the last century to rethink mathematics in terms of set theory was confronted, among other things, with the need to define every possible pair of coordinates as a particular type of set. This produced a difficulty of no small relevance. In fact, the order in which the elements of a set are listed does not matter. Therefore, {x, y} and {y, x} are exactly the same set. If we want a set in which the order of the elements does matter, we need something different. The challenge is that a set, every set, needs to be completely defined solely by its elements.

The first to find a satisfactory solution to the problem was the Polish mathematician Kazimierz Kuratowski in 1921. His definition of an ordered pair as a set of elements in which the order is unambiguously defined is still in use today:

In this definition, (x, y) is an abbreviation for the set {{x}, {x, y}}, in which {x, y} is an unordered pair of elements and {x} what establishes which of the two elements of the pair is the first.

An ordered pair is a set defined only by its elements

For this definition to determine the property we are interested in, that is, x is the first coordinate and y the second, there must be no other possible orderings. In particular, if the pair (x, y) is equal to another pair (u, v), it is necessary that x = u and y = v and that no other possibilities are given.

To prove that this is the case, once we have assumed that (x, y) = (u, v), we must analyze the consequences of the two possible alternatives, that is, when u = v and when, instead, u ≠ v.

If u = v, then

Hence,

This implies that

with the necessary consequence that x, y, u, and v are all the same, and, in particular, x = u and y = v.

The second alternative requires a little more complicated reasoning. If u ≠ v, then

Since (x, y) = (u, v), then

From the first statement it follows that {x} = {u} and, in turn, x = u. Given that {x} ≠ {u,v}, from the second statement it follows that

This implies that

It follows that x ≠ y and y ≠ u. Hence,

This proves that an ordered pair (x, y)={{x}, {x, y}} is a set whose order is unambiguously determined by its two elements only.

Cartesian product

If we have two sets A and B, we can form ordered pairs (x, y) with

The set of all ordered pairs formed in such a way is called the Cartesian product A × B of A and B. In symbols,

In plain language, the formula means that the cartesian product A × B is the set of all ordered pairs (x, y) such that x is an element of set A and y is an element of set B.

For example, let

Combining the elements of A and B to form ordered pairs, we get the following Cartesian product:

As usual, the order in which we list the elements of the set does not matter. Therefore, the following is the same as the previous set:

However, the order in which the elements of any ordered pair of the set are listed does matter. For example, (b, 0) is different from (0, b). Therefore, if we reverse the internal order of even one of the ordered pairs of the above set, it will no longer be the Cartesian product A × B of the sets A and B. In general, the Cartesian product of two sets A and B is not commutative unless A and B are equal or one of the two sets is empty.

Binary relations

A binary relation is a set whose elements are all ordered pairs. From this definition, it follows that the Cartesian product A × B of two sets A and B is a binary relation, since all its members are ordered pairs.

In general, R is a binary relation on a set A if and only if R is a subset of the Cartesian product A × A:

A binary relation (a, b) ∈ R is usually abbreviated as aRb. A typical example of a binary relation is the ordering relation <. For example, let R be the set of all ordered pairs (a, b) elements of ℕ such that a is less than b:

Since 1 is less than 2, we have (1, 2) ∈ R. This relation can be abbreviated as 1R2. Instead of R, we can use the symbol < to define the same subset of the Cartesian product ℕ × ℕ. Therefore, we can write concisely 1 < 2 meaning that the ordered pair (1, 2) is an element of the set <:

There are many types of binary relations. What distinguishes one type from another are the properties that hold for any relation. Some of these properties are mutually exclusive, as can easily be seen from the following list.

We say that a binary relation R on a set A is

• reflexive if for all a ∈ A, (a, a) ∈ R. An example of reflexive relation is the relation ‘is equal to’, applicable to any number since each number is equal to itself;
• symmetric if for all a, b ∈ A, (a, b) ∈ R implies (b, a) ∈ R. ‘John is Jack’s twin’ is a non-mathematical example of a symmetric binary relation, since it implies that ‘Jack is John’s twin’;
• transitive if for all a, b, c ∈ A, (a, b), (b, c) ∈ R implies (a, c) ∈ R. An example is the relation ‘is less than’. If 2 < 5 and 5 < 7, then 2 < 7.
• antireflexive if for all a ∈ A, (a, a) ∉ R. The binary relations ‘is greater than’ and ‘is less than’ are both antireflexive since no number can be greater or less than itself;
• antisymmetric if for all a, b ∈ A, (a, b) ∈ R and (b, a) ∈ R implies a = b. The divisibility relation on natural numbers is an example of an antisymmetric relation. For instance, four is divisible by two, but two is not divisible by four. The only way two natural numbers can be mutually divisible is if they are the same number: Four is divisible by four, two is divisible by two.

Ordered Pairs, Cartesian Product, Binary Relations was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.

One of the problems with natural languages is the ambiguity of many expressions. Nouns, verbs, and adjectives often have two or more meanings: Words that have a certain meaning in one context take on a completely different one in another context. For these and other reasons, fully mastering a language takes years of study. The subtleties of natural languages are so numerous and complex that, even with the advice of the best linguists, the largest IT companies on the planet have not yet been able to produce fully reliable machine translation software.

Thus, to avoid the problems caused by the ease with which natural languages produce ambiguous phrases, mathematics has long developed formal languages that have the express purpose of producing statements that are, as far as possible, precise, clear, concise, and, most of all, not self-contradictory.

It is perfectly possible to express a theorem and its proof in a natural language. Still, formal languages specifically designed to express mathematical concepts make the enunciation of theorems and proofs more precise and effective.

An exemplary case of formal language designed to eliminate any ambiguity is the language used in set theory, denoted by the symbol ℒ. This language uses few symbols to express many mathematical concepts. Since the twentieth century, almost all mathematics has been reformulated in terms of set theory. Therefore, knowing the basic elements of the set language allows us to have the necessary tools to understand the foundations on which modern mathematics is built.

The following are the basic symbols used in ℒ:

Using these symbols, it is possible to produce basic statements such as x = y, meaning x equals y, and x ∈ y, meaning x is an element of y.

Compound statements can be created by combining basic statements in many different ways. For example, we can write:

to say that b is an element of both x and y;

to say that x and y are different sets;

to say that the set y is empty;

to say that w is a subset of z;

to say that w has exactly one element. Basic and compound statements are also called formulas.

In addition, some abbreviations have been added to the language to make statements or formulas more concise. To say that x and y are different sets, the following two statements are perfectly equivalent:

Similarly, to say that b is not an element of y, we can write indifferently:

To say that the set y is empty, we can use any of the two following statements:

To say that w is a subset of z, we can write:

To say that w is a proper subset of z, i.e., that w is not equal to z, we can write:

instead of the more complex statement:

The expression

stands for the intersection of the sets x and y, i.e., the collection of all elements belonging to both x and y. The expression

means the union of sets x and y, i.e., the collection of all elements in either x or y. Finally, the expression

stands for the collection of all elements of x that are not in y.

This basic list of symbols is enough to understand and formulate even extremely complex statements expressed in the language of set theory.

ℒ, the Language of Set Theory was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.

Known since ancient times, magic squares are arrays filled with numbers, made up of an equal number of rows and columns. Numbers are usually positive integers and must all be distinct from each other. What makes these squares “magical” is that the sum of the numbers in each row, column, and main diagonal is always the same.

Some definitions

Magic squares are called normal if they are filled with a progressive series of natural numbers starting from 1 up to n², where n is the number of rows or columns that make up the square. The constant sum of the horizontal, vertical, and diagonal lines is called the magic constant and equals

For example, if the number of vertical lines is 4, the magic constant is

The number of rows and columns defines the order of the magic square. Therefore, a magic square with 5 rows and 5 columns will be of order 5, one with 6 rows and 6 columns of order 6, etc. Since the construction rules are different, it is also important to distinguish whether the number of rows and columns is odd or even. A magic square with an odd number of rows and columns is said to be of odd order; one with an even number is said to be of even order.

The smallest possible magic square has three rows and three columns. In fact, with two rows and two columns, that is, with four different numbers in all, it is not possible to construct one. And a magic square made of a single number would make the concepts of row, column, and diagonal meaningless.

The arrangement of the numbers in a magic square of order 3

We can easily determine the arrangement of the numbers in a magic square of order three with a little algebra and some calculation. Let us consider as a starting point a square filled with an ordered sequence of letters from a to i like the one shown in the following image.

By applying the formula for the magic constant to a square of order three, we get 15. Thus, we know that the numbers must be arranged in the square so that the sum of each row, column, and main diagonal is 15. Consequently, we also know that the sum of the two main diagonals and the median row and column must be 60.

Each letter appears only once in the four sums, except e, which, being in the center of the square, appears four times. Therefore, we can write the following equation:

Solving for e, we get:

Now, we know that the number in the center of the magic square must be 5. Furthermore, we can deduce the following equalities:

From these equalities, we derive that one of the complementary pairs of the central 5 must be 1 and 9. So let’s ask ourselves: Can 1 and 9 occupy one of the diagonals? The answer is no, as can be seen from the six test squares reproduced below.

In fact, if we put 1 and 9 in two opposite corners with 5 fixed in the center, none of the six remaining numbers could occupy one of the other corners of the square. 2, 3, and 4 cannot occupy a corner because they would be in the same row or column with 1, and, among the remaining numbers, there is none that, added to the other two, would give 15 as a sum. Conversely, 6, 7, and 8 cannot be in the same row or column with 9 because, in this case, the sum would be greater than 15.

There are also other reasons why we cannot put 1 and 9 in the corners of the square, but those listed are enough. Therefore, we conclude that 1 and 9 must be in the median row or column. The following four are the only possible positions for 1 and 9 in a normal magic square of order 3.

At this point, placing the remaining six numbers in the left empty cells is easy. Each choice is forced by the need to get 15 as the sum of each row, column, and main diagonal. We then obtain the following eight solutions, all equivalent to each other.

We can perform reflections and rotations on a magic square without altering the sums generated by the numbers it contains. By keeping the reciprocal positions of each number, we can get by reflections and rotations eight equivalent magic squares in all, as can be deduced from the solutions reported above for the magic square of order 3.

How to construct a magic square of odd order

The one of order 3 just examined is only a special case of a magic square of odd order.

There are several methods of generating odd-order magic squares. Probably the simplest and best known is the one also known as the “Siamese method,” attributed to the French diplomat and mathematician Simon de la Loubère. This method is based on the use of broken diagonals. So, before introducing his method, let’s see what a broken diagonal is and the difference between main diagonals and broken diagonals.

The main diagonals of a magic square are two, precisely those that start from one corner of the square and end in the opposite corner. They pass through all the rows and columns of the square in sequence.

On the other hand, a broken diagonal does not start from a corner but from an intermediate cell. Due to this, it reaches the opposite end of the square without crossing all the rows and columns but only part of them. It is therefore broken into two parts. The second piece of the diagonal begins to cross the square starting from the first row/column not crossed by the first piece of the diagonal. To be complete, a broken diagonal must cross each row/column of the magic square once.

There are two types of broken diagonals, one in which the numbers grow downward, one in which the numbers grow upward. The following image illustrates the two types of broken diagonal.

The first step in creating an odd-order magic square with de la Loubère’s method is to put 1 in the upper central cell of the square.

The second step is to put 2 in the first available cell of the upward broken diagonal to which the cell where we put 1 belongs. At this point, we complete the broken diagonal by inserting the numbers following 2.

Once the diagonal is complete, we enter the next number in the cell below the one in which we entered the last number of the previous diagonal. Therefore, we complete in the same way the second broken diagonal and all subsequent ones, depending on the size of the magic square, always inserting the first number of each new diagonal in the cell below the one in which we entered the last number of the previous diagonal.

Below is a magic square of order 5 created with de la Loubère’s method. The magic constant is 65. The different colors help to visualize the broken diagonals that make up the magic square.

Below a magic square of order 7 always created with de la Loubère’s method. The magic constant is 175.

How to construct a magic square of even order

There are also several methods for constructing magic squares of even order. One of the most interesting dates back to the early 1900s and is attributed to a certain Mr. Devedec about whom absolutely nothing is known. But his method works.

The construction of a magic square with the Devedec method is based on a scheme that uses four symbols to indicate which number should be inserted in each square position. The symbols refer to the positions of the numbers in a fundamental square, in which all numbers are entered in progressive order from left to right and top to bottom.

The four symbols are the following:

• The circle indicates a cell that we must fill with a number that is in the same position in the fundamental square;
• the diagonal line indicates a cell that we must fill with a number that is in a symmetrical position to the center in the fundamental square;
• the vertical line indicates a cell that we must fill with a number that is in a symmetrical position to the horizontal median line in the fundamental square;
• the horizontal line indicates a cell that we must fill with a number that is in a symmetrical position to the vertical median line in the fundamental square.

Below is the scheme of the symbols for a magic square of order 8 made by Devedec’s method. The different colors help to better perceive the regularities in the distribution of symbols.

The one below is a magic square of order 8 made with the Devedec scheme shown above. The sum of the numbers in each row, column, and main diagonal is 260.

Comparing the numbers in this magic square with the scheme shown above and with the arrangement of the numbers in the fundamental square, it is easy to see that the numbers on a yellow background, corresponding to the symbol of the circle, are in the same position as they are in the fundamental square. Instead, the numbers on an azure background, corresponding to the horizontal line symbol, are in a symmetrical position to the vertical median line of the square. For example, 6 is in the position of 3, and 3 is in the position of 6.

Analogous displacements based on the symmetries associated with the other two symbols are valid for the remaining numbers of the square. Once one has learned the “grammar” of the symmetries of the Devedec scheme, filling the magic square with the appropriate numbers requires only a little patience and attention.

Below, the filling pattern of a magic square of order 10 and the magic square itself are shown. The magic constant is 505.

Before concluding this review of magic squares of even order, it is important to highlight a detail. As you can see by comparing the filling pattern of a magic square of order 8 with that of order 10, there are in both eight central boxes, filled with different symbols but all marked with a single color.
The Devedec method requires that those eight central boxes have a different pattern, depending on whether the number of rows/columns of the square is a multiple of 4 or not. If the order is a multiple of 4, as in the case of the magic square of order 8, then we have the central pattern with the circles. On the other hand, if the order is not a multiple of 4, as in the case of the magic square of order 10, then the scheme of the central squares does not contain circles.

The difference between the two central patterns is shown in the following image, which shows side by side the two general patterns to create magic squares of order 8 and 10.

For the rest, there are no other differences. It is possible to make magic squares of even larger order than those shown here simply by expanding the general scheme given above. Just take care to respect the various symmetries in the distribution of the symbols that can be inferred from the magic squares presented in this paragraph.

Magic Squares and How to Construct them was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.

The power set of a set X, denoted 𝒫(X), is the set of all subsets of X, including X itself and the empty set.

Let us begin with the smallest of all sets, the empty set. If

then

A little more complicated case is a set with only one element. If

then

Now, let us examine the cases in which X has two, three, and four elements, respectively:

We have listed until now five sets, increasing by one the number of elements at each step. Counting the number of elements in each set and the relative power sets, we can notice the following relation:

We can derive a general rule from this short list: If n is the number of elements in a set, 2ⁿ is the number of elements in its power set. In other words, every time we increase by one the number of elements in a set, the number of subsets in its power set doubles.

Why is it so? The reason is simple. The power set of a set containing n+1 elements contains all the subsets already listed in the power set of a set containing n elements plus all the subsets created by adding the new element. As it is easy to verify looking at the examples above, the number of subsets created just by adding this new element equals the number of subsets in the power set of the set with n elements. The obvious consequence is that the number of elements in a power set doubles every time we increase by 1 the elements in the starting set.

The following are the power sets of X when X has three and four elements, respectively. The subsets of 𝒫(X) containing the new element d are highlighted in red to immediately clarify which subsets it was necessary to add to the previous power set.

Power Set was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.

A set is a collection of objects. It can be thought of as the container of those objects and, together, as what gives them unity.

Membership, order, listing

The objects that belong to a set are called elements or members of the set. The symbol ∈ indicates that an object belongs to a set. For example, to say that the object x belongs to the set A, we write:

Instead, to indicate that x does not belong to A, we write:

For sets consisting of a few elements, it is convenient to list the elements of the set. Usually, the list of elements, separated by commas, is enclosed in curly brackets and preceded by the set’s name, denoted by a capital letter, and by the equal sign. For example, if A is the set of perfect squares less than 50, then we will write:

The order in which the elements are listed does not matter. The following two writings are completely equivalent:

Furthermore, each element must appear only once in the listing. For example, the set of the consonants of the word ‘shuffle’ is {s, h, f, l}, not {s, h, f, f, l}.

Principle of extensionality

A fundamental aspect of set theory is that a set is completely determined by its elements. To understand what this means, let’s define A as the set of the consonants of the word ‘shuffle’ and B as the set of the consonants of the word ‘shelf.’ The two sets will therefore be:

As can be seen, they contain precisely the same four elements s, h, l, f. Therefore, we say that A = B. It doesn’t matter that the two sets have been defined differently and that the order in which the elements are listed is slightly different. What matters is that both sets have exactly the same elements and that there is no ambiguity about it.

Saying that two sets are equal means that they are the same object. What is true for the first is equally true for the second.

In set theory, the fact that the identity of two sets is determined exclusively by their elements is called the Principle of Extensionality. It states that if two sets have exactly the same elements, then they are equal. In symbols, this property can be synthetically expressed, for any object x, as follows:

The expression means: If x is an element of A necessarily implies that x is an element of B, then A = B.

Conversely, if A = B, then any object x, if it belongs to A, necessarily also belongs to B:

If two sets A and B are not equal, we will write:

The empty set

The set A listed below contains only 0 as its element:

However, there is an even smaller set than a set that contains only one element. It is the empty set, which, as the name implies, does not contain any elements. It is denoted by the symbol ∅. Having no elements, it is the only set that we can fully describe simply using one symbol, namely ∅. Moreover, by the extensionality principle, the empty set is unique. In fact, all empty sets are necessarily equal and, as such, are the same set.

Although the empty set does not contain any elements, it can be the content of another set:

In this case, A is the set that contains the empty set. In this regard, it is important to note that the set containing the empty set, {∅}, is different from the empty set itself ∅. In symbols, we will write:

The difference between the two cases is the same as between a person who has an empty container and one who does not even have the container. This difference becomes clearer if we reflect on the fact that the empty set is an element — the only element — of the set that contains the empty set:

Conversely, the empty set is not an element of the empty set:

Indeed, if it were, the empty set would no longer be empty.

Sets of sets

A set can contain other sets, creating a layering of contents. For example, the set containing the set whose only element is the empty set is:

Similarly, the set of the set that contains the set whose only element is the empty set is:

The following set contains three elements, namely the empty set, the set containing the empty set, and the set containing the set whose only element is the empty set:

Union of sets

The union of two sets A and B, denoted by the symbol ∪, is the set that contains all the elements of A and all the elements of B. Here are some examples:

Intersection of sets

The intersection of two sets A and B, denoted by the symbol ∩, is the set that contains only the elements common to A and B. Here are some examples:

Two sets whose intersection is the empty set, that is, two sets that have no element in common, are said to be disjoint.

Subsets

A set A is a subset of the set B if all the elements of A are also elements of B. In symbols,

This relation means that A is included in B or, symmetrically, that B includes A. Each set is a subset of itself. Hence, for any set A, it is always true that:

A special case is the empty set, which is a subset of any set, including itself:

However, we must not confuse the membership relation of an element to a set defined by the symbol ∈ with the inclusion relation of a subset in a set defined by the symbol ⊆. For example, while it is true that the empty set is a subset of itself:

it is not true that the empty set is an element of itself:

Unlike the previous case, the set containing the empty set is an element of the set containing the set whose only element is the empty set:

However, the first set is not a subset of the second:

It is not a subset of the second set because the empty set is an element of the first set, not the second. In fact, the only element of the second set is the set that contains the empty set, but not the empty set itself. (Recall that, for A to be a subset of B, all the elements of A must also be in B.)

Basics of Sets was originally published in Not Zero on Medium, where people are continuing the conversation by highlighting and responding to this story.