In Chapter 1 of RD Sharma's Class 11 textbook Sets are one of the foundational topics of mathematics. Exercise 1.3 dives deeper into operations on sets such as the union, intersection, and difference helping students understand these concepts through a variety of problems. This exercise builds a strong base for understanding more advanced mathematical concepts in later chapters.
What is a Set?
A set is a well-defined collection of distinct objects considered as an object in its own right. The objects in a set are called elements or members. The Sets are typically denoted by the capital letters like A, B, and C and are represented in curly braces {}. For example, A={1,2,3} represents a set of the three elements. The Sets can be finite, infinite, or empty and are used to describe collections of numbers, objects, or symbols in mathematics.
Question 1. Which of the following are examples of the empty set?
(i) Set of all even natural numbers divisible by 5.
(ii) Set of all even prime numbers.
(iii) {x: x2–2=0 and x is rational}.
(iv) {x: x is a natural number, x < 8 and simultaneously x > 12}.
(v) {x: x is a point common to any two parallel lines}.
Solution:
(i) All the numbers ending with 0. Except 0 is divisible by 5 and are even natural number.
Therefore, it is not an example of empty set.
(ii) 2 is a prime number and is even, and it is the only prime which is even.
Therefore, this not an example of the empty set.
(iii) x2 – 2 = 0, x2 = 2, x = ± √2 ∈ N. There is not natural number whose square is 2.
Therefore, it is an example of empty set.
(iv) There is no natural number less than 8 and greater than 12.
Therefore, it is an example of the empty set.
(v) No two parallel lines intersect at each other.
Therefore, it is an example of empty set.
Question 2. Which of the following sets are finite and which are infinite?
(i) Set of concentric circles in a plane.
(ii) Set of letters of the English Alphabets.
(iii) {x ∈ N: x > 5}
(iv) {x ∈ N: x < 200}
(v) {x ∈ Z: x < 5}
(vi) {x ∈ R: 0 < x < 1}.
Solution:
(i) Infinite concentric circles can be drawn in a plane.
Therefore, it is an infinite set.
(ii) There are just 26 letters in English Alphabets.
Therefore, it is finite set.
(iii) It is an infinite set because, natural numbers greater than 5 is infinite.
(iv) It is a finite set. Since, natural numbers start from 1 and there are 199 numbers less than 200.
Therefore, it is a finite set.
(v) It is an infinite set. Because integers less than 5 are infinite.
Therefore, it is an infinite set.
(vi) It is an infinite set. Because between two real numbers, there are infinite real numbers.
Question 3. Which of the following sets are equal?
(i) A = {1, 2, 3}
(ii) B = {x ∈ R: x2–2x+1=0}
(iii) C = (1, 2, 2, 3}
(iv) D = {x ∈ R : x3 – 6x2+11x – 6 = 0}
Solution:
A set is said to be equal with another set if all elements of both the sets are equal and same.
A = {1, 2, 3}
B ={x ∈ R: x2 – 2x+1=0}
x2 – 2x+1 = 0
(x–1)2 = 0
Therefore, x = 1.
B = {1}
C= {1, 2, 2, 3}
In sets we do not repeat elements hence C can be written as {1, 2, 3}
D = {x ∈ R: x3 – 6x2+11x – 6 = 0}
For x = 1, x2–2x+1=0
= (1)3–6(1)2+11(1)–6
= 1–6+11–6
= 0
For x =2,
= (2)3–6(2)2+11(2)–6
= 8–24+22–6
= 0
For x =3,
= (3)3 – 6(3)2+11(3)–6
= 27–54+33–6
= 0
Therefore, D = {1, 2, 3}
Hence, the set A, C and D are equal.
Question 4. Are the following sets equal?
A={x: x is a letter in the word reap},
B={x: x is a letter in the word paper},
C={x: x is a letter in the word rope}.
Solution:
For A
Letters in word reap
A ={R, E, A, P} = {A, E, P, R}
For B
Letters in word paper
B = {P, A, E, R} = {A, E, P, R}
For C
Letters in word rope
C = {R, O, P, E} = {E, O, P, R}.
Set A = Set B
Because every element of set A is present in set B
But Set C is not equal to either of them because all elements are not present.
Question 5. From the sets given below, pair the equivalent sets:
A= {1, 2, 3}, B = {t, p, q, r, s}, C = {α, β, γ}, D = {a, e, i, o, u}.
Solution:
Equivalent set are different from equal sets, Equivalent sets are those which have equal number of elements they do not have to be same.
A = {1, 2, 3}
Number of elements = 3
B = {t, p, q, r, s}
Number of elements = 5
C = {α, β, γ}
Number of elements = 3
D = {a, e, i, o, u}
Number of elements = 5
Therefore, Set A is equivalent with Set C.
Set B is equivalent with Set D.
Question 6. Are the following pairs of sets equal? Give reasons.
(i) A = {2, 3}, B = {x: x is a solution of x2 + 5x + 6= 0}
(ii) A={x : x is a letter of the word “WOLF”}
B={x : x is letter of word “FOLLOW”}
Solution:
(i) A = {2, 3}
B = x2 + 5x + 6 = 0
x2 + 3x + 2x + 6 = 0
x(x+3) + 2(x+3) = 0
(x+3) (x+2) = 0
x = -2 and -3
= {–2, –3}
Since, A and B do not have exactly same elements hence they are not equal.
(ii) Every letter in WOLF
A = {W, O, L, F} = {F, L, O, W}
Every letter in FOLLOW
B = {F, O, L, W} = {F, L, O, W}
Therefore, A and B have same number of elements which are exactly same, hence they are equal sets.
Question 7. From the sets given below, select equal sets and equivalent sets.
A = {0, a}, B = {1, 2, 3, 4}, C = {4, 8, 12},
D = {3, 1, 2, 4}, E = {1, 0}, F = {8, 4, 12},
G = {1, 5, 7, 11}, H = {a, b}
Solution:
A = {0, a}
B = {1, 2, 3, 4}
C = {4, 8, 12}
D = {3, 1, 2, 4} = {1, 2, 3, 4}
E = {1, 0}
F = {8, 4, 12} = {4, 8, 12}
G = {1, 5, 7, 11}
H = {a, b}
Equivalent sets:
i. A, E, H (They have exactly two elements in them)
ii. B, D, G (They have exactly four elements in them)
iii. C, F (They have exactly three elements in them)
Equal sets:
i. B, D (all of them have exactly the same elements, Hence they are equal)
ii. C, F (all of them have exactly the same elements, Hence they are equal)
Question 8. Which of the following sets are equal?
A = {x : x ∈ N, x < 3}
B = {1, 2}, C= {3, 1}
D = {x : x ∈ N, x is odd, x < 5}
E = {1, 2, 1, 1}
F = {1, 1, 3}
Solution:
A = {1, 2}
B = {1, 2}
C = {3, 1}
D = {1, 3} (since, the odd natural numbers less than 5 are 1 and 3)
E = {1, 2} (since, repetition is not allowed)
F = {1, 3} (since, repetition is not allowed)
Therefore, Sets A, B and E are equal.
C, D and F are equal.
Question 9. Show that the set of letters needed to spell “CATARACT” and the set of letters needed to spell “TRACT” are equal.
Solution:
For “CATARACT”
Distinct letters are
{C, A, T, R} = {A, C, R, T}
For “TRACT”
Distinct letters are
{T, R, A, C} = {A, C, R, T}
As it is seen that the letters need to spell cataract is equal to set of letters need to spell tract.
Therefore, the two sets are equal.
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Summary
Exercise 1.3 covers set difference and symmetric difference. Students learn to find the difference and symmetric difference of two sets. Set difference is used to find elements in one set but not in another. Symmetric difference is used to find elements in one or the other set but not in both. Understanding set difference and symmetric difference is crucial for advanced math concepts.