Two sets are said to be equal if they contain the same elements, regardless of the order in which they are listed. In other words, sets A and B are equal if every element of set A is also an element of set B, and vice versa.
Example: P = {a, b, c, d} and Q = {b, a, d, c} are equal sets since they both have the same elements and also the same number of elements.
Equal Set Symbol
Equal Set is represented by the "=" sign between any two sets, where the equality holds. For example, {2, 3, 5} and {3, 2, 5} are equal sets and we can represent them using "=" symbol as follows:
{2, 3, 5} = {3, 2, 5}
Unlike this, unequal sets are represented by "≠", which means the equality between sets doesn't hold. For example,
{2, 3, 5} ≠ {1, 2, 3}
Example of Equal Sets
Let P be the set of all integers greater than 0, and let Q be the set of all natural numbers.
As we can see, all elements of P are the same as all the elements of Q; P and Q are equal sets.
Some other examples include:
- A = { 1, 2, 3, 4, 5} and B = {2, 3, 1, 5, 4}.
- Set of alphabets in words "listen" and "silent".
- Set of fractions {1/2, 2/4, 3/6} and {6/12, 4/8, 2/4}.
Equal vs Unequal Sets
The key differences between both equal and unequal sets are as follows:
| Equal Sets | Unequal Sets |
|---|---|
| Two sets have the same elements. | Two sets have different elements. |
| A = B | A ≠ B |
| Equal | May or May not be Equal |
| {1, 2, 3} = {3, 2, 1} | {1, 2, 3} ≠ {4, 5, 6} |
| Every subset of A is also a subset of B, and vice versa. | Subsets may differ. |
| A ∩ B = A (or B) | A ∩ B has common elements of both A and B. |
| A ∪ B = A (or B) | A ∪ B combines elements of both A and B. |
| Complement of A is the same as the complement of B. | Complements of unequal sets differ. |
Equal vs Equivalent Sets
Equal and equivalent sets are used to compare sets, visually, it can be represented as:
The key differences between equivalent and equal sets are given in the following table:
Equal Sets | Equivalent Sets |
|---|---|
Two or more sets are equal when all their elements are equal. | Two or more elements are equivalent when they have the same number of elements. |
Equal sets are denoted by the symbol '='. | Equivalent sets are denoted by the symbol '~' or '≡'. |
Equal sets are a broader term and encompass equivalent sets, i.e., all equal sets are also equivalent sets. | Two or more equivalent sets may or may not be equal. |
All elements of equal sets need to be the same. | The elements of two equivalent sets need not be the same. |
Note: Equal Sets are always Equivalent Sets, but vice versa is not true.
Venn Diagram of Equal Sets
The following Venn diagram shows set A = {2, 3, 5} = set B.
Properties of Equal Sets
There are various properties of equal sets, some of which are listed as follows:
- The intersection of two equal sets is equal to both sets, i.e., if A = B then, A ∩ B = A = B.
- Two equal sets are always subsets of each other, i.e., if A ⊂ B and B ⊂ A, then A = B.
- For two sets to be equal, the order of their elements does not matter, i.e., {9, 10, 11} = {11, 10, 9}.
- The cardinality of equal sets and their power set are the same.
- Equal sets always have the same number of elements.
- The elements of two equal sets are equal.
Solved Example on Equal Sets
Question 1: Are the sets P = {r: r is prime such that 40 < r < 50} and Q = {42, 44, 45, 46, 48} equal?
Solution:
Set P = {r: r is prime such that 40 < r < 50} and set Q = {42, 44, 45, 46, 48, 49}.
Thus, P = set of prime numbers between 40 and 50.
⇒ P = {41, 43, 47} ≠ {42, 44, 45, 46, 48, 49} = Q
Thus, sets P and Q are unequal.
Question 2: Identify the equal sets from the following:
- P = {p ∈ R: p2– 2p + 1 = 0}
- Q = (1, 2, 3}
- R = {p ∈ R : p3 – 6p2 + 11p – 6 = 0}.
Solution:
Two sets are regarded as equal sets when they have all the same elements and also the same number of elements.
Let's list out the elements of sets P and R before comparing them with set Q.
P ={p ∈ R: p2 – 2p + 1 = 0}
⇒ p2 – 2p + 1 = 0
⇒ (p - 1)2 = 0
∴ p = 1.⇒ P = {1}
Set Q can also be written as {1, 2, 3} since we do not repeat elements in a set.
Similarly, upon solving p3 – 6p2 + 11p – 6 = 0, set R = {1, 2, 3}.Thus, sets Q and R are equal.
Question 3: Determine the groups of equivalent and equal sets from the following: A = {0, $}, B = {10, 21, 39, 94}, C = {44, 89, 128}, D = {39, 10, 21, 94}, E = {1, 0}, F = {89, 44, 128}, G = {15, 5, @, 11}, H = {a, c}.
Solution:
Equivalent Sets:
Having 2 elements each: A, E and H
Having 3 elements each: C and F
Having 4 elements each: B, D and GEqual Sets:
B and D = {10, 21, 39, 94}
C and F = {44, 89, 128}
Question 4: Determine whether the sets of alphabets in words TITLE and LITTLE are equal.
Solution:
Let A be the set of alphabets in the word TITLE.
A = {L, I, T, E}Let B be the set of alphabets in the word LITTLE.
B = {L, I, T, E}Thus, A and B are equal sets.
Question 5: Check if Set A = {2,4,6,8} and Set B = {x:x is positive integer less than 10} are equal sets or not.
Solution:
Set A = {2,4,6,8}
This set contains the elements 2,4,6 and 8.Set B = {x:x is a positive inteer less than 10}
This setis divided as the set of all positive interger less than 10. The positive even integer less than 10 are 2,4,6 and 8.Since both sets have exactly the same elements A = B.
Practice Problems of Equal Sets
Question 1. Determine if A = {a, b, c} and {b, c, a} are equal set or not.
Question 2. Check if set A = {2, 4, 6, 8} and set B = (x: x is positive even integer less then 10)
Question 3. Determine if the sets P = (x: x is roots of equation, x2 + 5x + 6 = 0) and Q = {2, 3} are equal set or not.
Question 4: Check if the sets A = {1, 3, 5, 7} and B = {x: x is an odd number less than 8} are equal sets or not.
Question 5: Determine if X = {2, 4, 6} and Y = {1 + x: x is a positive even integer less than 6} are equal sets or not.