Exercise 1.1
Which of the following are sets? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-cricket batsmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
Answer
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.
Ans.
(i) Yes, this is a set. The months beginning with J are January, June, and July — a well-defined, definite collection.
(ii) No. "Most talented" is subjective; different people may have different opinions. The collection is not well-defined.
(iii) No. "Best" is subjective and varies from person to person. Not a well-defined collection.
(iv) Yes, this is a set. The boys in a particular class are clearly identifiable.
(v) Yes, this is a set. Any natural number can be tested to determine if it is less than 100.
(vi) Yes, this is a set. The novels written by Munshi Prem Chand are a definite, identifiable collection.
(vii) Yes, this is a set. Any integer can be tested to see if it is even.
(viii) Yes, this is a set. The questions in the chapter are a definite collection.
(ix) No. "Most dangerous" is subjective; there is no single agreed-upon criterion.
Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces:
Answer
(i) 5 ___ A (ii) 8 ___ A (iii) 0 ___ A (iv) 4 ___ A (v) 2 ___ A (vi) 10 ___ A
Ans.
(i) 5 ∈ A (ii) 8 ∉ A (iii) 0 ∉ A (iv) 4 ∈ A (v) 2 ∈ A (vi) 10 ∉ A
Q3. Write the following sets in roster form
(i) A = {x : x is an integer and –3 ≤ x < 7}
(ii) B = {x : x is a natural number less than 6}
(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}
(iv) D = {x : x is a prime number which is divisor of 60}
(v) E = The set of all letters in the word TRIGONOMETRY
(vi) F = The set of all letters in the word BETTER
Ans.
(i) A = {–3, –2, –1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {1, 2, 3, 4, 5}
(iii) C = {17, 26, 35, 44, 53, 62, 71, 80}
(iv) 60 = 2 × 2 × 3 × 5; prime divisors are 2, 3, 5. D = {2, 3, 5}
(v) E = {T, R, I, G, O, N, M, E, Y}
(vi) F = {B, E, T, R}
Q4. Write the following sets in the set-builder form
(i) {3, 6, 9, 12} (ii) {2, 4, 8, 16, 32} (iii) {5, 25, 125, 625} (iv) {2, 4, 6, …} (v) {1, 4, 9, …, 100}
Ans.
(i) {x : x = 3n, n ∈ N, 1 ≤ n ≤ 4}
(ii) {x : x = 2n, n ∈ N, 1 ≤ n ≤ 5}
(iii) {x : x = 5n, n ∈ N, 1 ≤ n ≤ 4}
(iv) {x : x = 2n, n ∈ N}
(v) {x : x = n², n ∈ N, 1 ≤ n ≤ 10}
Q5. List all the elements of the following sets
(i) A = {x : x is an odd natural number}
(ii) B = {x : x is an integer, −1/2 < x < 9/2}
(iii) C = {x : x is an integer, x² ≤ 4}
(iv) D = {x : x is a letter in the word "LOYAL"}
(v) E = {x : x is a month of a year not having 31 days}
(vi) F = {x : x is a consonant in the English alphabet which precedes k}
Ans.
(i) A = {1, 3, 5, 7, 9, …}
(ii) −1/2 = −0.5 and 9/2 = 4.5; integers in this range: B = {0, 1, 2, 3, 4}
(iii) x² ≤ 4 means −2 ≤ x ≤ 2; C = {−2, −1, 0, 1, 2}
(iv) D = {L, O, Y, A}
(v) E = {February, April, June, September, November}
(vi) F = {b, c, d, f, g, h, j}
Match each of the sets on the left in roster form with the same set on the right in set-builder form:
(i) {1, 2, 3, 6} — (a) {x : x is a prime number and a divisor of 6}
(ii) {2, 3} — (b) {x : x is an odd natural number less than 10}
(iii) {M,A,T,H,E,I,C,S} — (c) {x : x is a natural number and divisor of 6}
(iv) {1, 3, 5, 7, 9} — (d) {x : x is a letter of the word MATHEMATICS}
Answer
(i)→(c), (ii)→(a), (iii)→(d), (iv)→(b)
Exercise 1.2
Which of the following are examples of the null set?
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) {x : x is a natural number, x < 5 and x > 7}
(iv) {y : y is a point common to any two parallel lines}
Answer
(i) Null set — no odd number is divisible by 2.
(ii) Not null — {2} is the set of even primes.
(iii) Null set — no number can be simultaneously less than 5 and greater than 7.
(iv) Null set — parallel lines never intersect.
Which of the following sets are finite or infinite?
(i) Set of months of a year (ii) {1, 2, 3, …} (iii) {1, 2, 3, …, 99, 100}
(iv) Positive integers greater than 100 (v) Prime numbers less than 99
Answer
(i) Finite (12 months) (ii) Infinite (iii) Finite (100 elements) (iv) Infinite (v) Finite (25 primes)
Q3. State whether each of the following set is finite or infinite
(i) Lines parallel to the x-axis (ii) Letters in the English alphabet (iii) Multiples of 5
(iv) Animals living on the earth (v) Circles passing through origin (0, 0)
Ans. (i) Infinite (ii) Finite (26) (iii) Infinite (iv) Finite (v) Infinite
Q4. In the following, state whether A = B or not
(i) A = {a, b, c, d}, B = {d, c, b, a} (ii) A = {4, 8, 12, 16}, B = {8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10}, B = {x : x is a positive even integer and x ≤ 10}
(iv) A = {x : x is a multiple of 10}, B = {10, 15, 20, 25, 30, …}
Ans.
(i) A = B (same elements, order immaterial)
(ii) A ≠ B (A has 12; B has 18 instead)
(iii) A = B (both equal {2, 4, 6, 8, 10})
(iv) A ≠ B (B includes 15, 25, etc., which are not multiples of 10)
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 in FOLLOW}, B = {y : y is a letter in WOLF}
Answer
(i) x² + 5x + 6 = 0
⇒ x = −2 or −3; B = {−2, −3} ≠ {2, 3} = A. Not equal.
(ii) Both give {F, O, L, W}. A = B.
Q6. From the sets given below, select equal sets
A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2}, E = {−1, 1}, F = {0, a}, G = {1, −1}, H = {0, 1}
Ans. B = D (both contain 1, 2, 3, 4) and E = G (both contain 1 and −1).
Exercise 1.3
Q1. Fill in ⊂ or ⊄
(i) {2,3,4} ___ {1,2,3,4,5} (ii) {a,b,c} ___ {b,c,d} (iii) {Class XI students} ___ {school students}
(iv) {circles in plane} ___ {circles with radius 1} (v) {triangles} ___ {rectangles}
(vi) {equilateral triangles} ___ {triangles} (vii) {even natural numbers} ___ {integers}
Ans. (i) ⊂ (ii) ⊄ (iii) ⊂ (iv) ⊄ (v) ⊄ (vi) ⊂ (vii) ⊂
Q2. Examine whether true or false
(i) {a, b} ⊄ {b, c, a} (ii) {a, e} ⊂ {vowels} (iii) {1,2,3} ⊂ {1,3,5}
(iv) {a} ⊂ {a,b,c} (v) {a} ∈ {a,b,c} (vi) {even naturals < 6} ⊂ {divisors of 36}
Ans. (i) False — {a,b} ⊂ {b,c,a} (ii) True (iii) False (2 ∉ {1,3,5}) (iv) True (v) False (vi) True (2 and 4 both divide 36)
Let A = {1, 2, {3, 4}, 5}. Which statements are incorrect and why?
(i) {3,4} ⊂ A (ii) {3,4} ∈ A (iii) {{3,4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A
(vi) {1,2,5} ⊂ A (vii) {1,2,5} ∈ A (viii) {1,2,3} ⊂ A (ix) φ ∈ A (x) φ ⊂ A (xi) {φ} ⊂ A
Answer
Elements of A are 1, 2, {3,4}, 5.
(i) Incorrect — {3,4} is an element of A, not a subset; 3 and 4 individually are not in A.
(ii) Correct — {3,4} ∈ A.
(iii) Correct — {{3,4}} ⊂ A since {3,4} is an element of A.
(iv) Correct.
(v) Incorrect — 1 is an element, not a set; ⊂ is used between sets.
(vi) Correct — 1, 2, 5 ∈ A.
(vii) Incorrect — {1,2,5} is a subset of A, not an element.
(viii) Incorrect — 3 ∉ A (only {3,4} is in A).
(ix) Incorrect — φ is not an element listed in A.
(x) Correct — φ ⊂ every set.
(xi) Incorrect — φ is not an element of A so {φ} ⊄ A.
Q4. Write down all the subsets of
(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ
Ans.
(i) φ, {a}
(ii) φ, {a}, {b}, {a,b}
(iii) φ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
(iv) φ
Q5. Write as intervals
(i) {x : x ∈ R, −4 < x ≤ 6} (ii) {x : x ∈ R, −12 < x < −10}
(iii) {x : x ∈ R, 0 ≤ x < 7} (iv) {x : x ∈ R, 3 ≤ x ≤ 4}
Ans. (i) (−4, 6] (ii) (−12, −10) (iii) [0, 7) (iv) [3, 4]
Q6. Write as set-builder form
(i) (−3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [−23, 5)
Ans.
(i) {x : x ∈ R, −3 < x < 0}
(ii) {x : x ∈ R, 6 ≤ x ≤ 12}
(iii) {x : x ∈ R, 6 < x ≤ 12}
(iv) {x : x ∈ R, −23 ≤ x < 5}
Q7. What universal set(s) would you propose for
(i) The set of right triangles. (ii) The set of isosceles triangles.
Ans. Both: the set of all triangles in a plane (or the set of all polygons).
Given A = {1,3,5}, B = {2,4,6}, C = {0,2,4,6,8}; which can be universal set for all three?
Answer
(i) {0,1,2,3,4,5,6} (ii) φ (iii) {0,1,2,3,4,5,6,7,8,9,10} (iv) {1,2,3,4,5,6,7,8}
Ans. Only (iii). Option (i) lacks 8 (needed for C); (ii) is empty; (iv) lacks 0 (needed for C).
Exercise 1.4
Q1. Find the union of each pair of sets
(i) X = {1,3,5}, Y = {1,2,3} (ii) A = {a,e,i,o,u}, B = {a,b,c}
(iii) A = {multiples of 3}, B = {naturals < 6} (iv) A = {x : 1 < x ≤ 6}, B = {x : 6 < x < 10}
(v) A = {1,2,3}, B = φ
Ans.
(i) X ∪ Y = {1, 2, 3, 5}
(ii) A ∪ B = {a, b, c, e, i, o, u}
(iii) A ∪ B = {1, 2, 3, 4, 5, 6, 9, 12, 15, …}
(iv) A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}
(v) A ∪ B = {1, 2, 3}
Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?
Answer
Yes, A ⊂ B. A ∪ B = {a, b, c} = B.
If A ⊂ B, what is A ∪ B?
Answer
A ∪ B = B.
A = {1,2,3,4}, B = {3,4,5,6}, C = {5,6,7,8}, D = {7,8,9,10}. Find:
Answer
(i) A ∪ B (ii) A ∪ C (iii) B ∪ C (iv) B ∪ D (v) A ∪ B ∪ C (vi) A ∪ B ∪ D (vii) B ∪ C ∪ D
Ans.
(i) {1,2,3,4,5,6} (ii) {1,2,3,4,5,6,7,8} (iii) {3,4,5,6,7,8} (iv) {3,4,5,6,7,8,9,10}
(v) {1,2,3,4,5,6,7,8} (vi) {1,2,3,4,5,6,7,8,9,10} (vii) {3,4,5,6,7,8,9,10}
Find the intersection of each pair in Q1.
Answer
(i) {1,3} (ii) {a} (iii) {3} (iv) φ (v) φ
A = {3,5,7,9,11}, B = {7,9,11,13}, C = {11,13,15}, D = {15,17}. Find:
Answer
(i) A ∩ B (ii) B ∩ C (iii) A ∩ C ∩ D (iv) A ∩ C (v) B ∩ D (vi) A ∩ (B ∪ C)
(vii) A ∩ D (viii) A ∩ (B ∪ D) (ix) (A ∩ B) ∩ (B ∪ C) (x) (A ∪ D) ∩ (B ∪ C)
Ans.
(i) {7,9,11} (ii) {11,13} (iii) φ (iv) {11} (v) φ
(vi) B ∪ C = {7,9,11,13,15}; A ∩ (B ∪ C) = {7,9,11}
(vii) φ (viii) B ∪ D = {7,9,11,13,15,17}; A ∩ (B ∪ D) = {7,9,11}
(ix) A ∩ B = {7,9,11}; ∩ (B ∪ C) = {7,9,11}
(x) A ∪ D = {3,5,7,9,11,15,17}; B ∪ C = {7,9,11,13,15}; intersection = {7,9,11,15}
A = naturals, B = even naturals, C = odd naturals, D = primes. Find:
Answer
(i) A ∩ B (ii) A ∩ C (iii) A ∩ D (iv) B ∩ C (v) B ∩ D (vi) C ∩ D
Ans.
(i) B (ii) C (iii) D (iv) φ (v) {2} (vi) {3,5,7,11,…} (odd primes)
Which pairs of sets are disjoint?
Answer
(i) {1,2,3,4} and {4,5,6} (ii) {a,e,i,o,u} and {c,d,e,f} (iii) even integers and odd integers
Ans. (i) Not disjoint (4 is common) (ii) Not disjoint (e is common) (iii) Disjoint
A = {3,6,9,12,15,18,21}, B = {4,8,12,16,20}, C = {2,4,6,8,10,12,14,16}, D = {5,10,15,20}. Find:
Answer
(i) A−B (ii) A−C (iii) A−D (iv) B−A (v) C−A (vi) D−A
(vii) B−C (viii) B−D (ix) C−B (x) D−B (xi) C−D (xii) D−C
Ans.
(i) {3,6,9,15,18,21} (ii) {3,9,15,21} (iii) {3,6,9,12,18,21} (iv) {4,8,16,20}
(v) {2,4,8,10,14,16} (vi) {5,10,20} (vii) {20} (viii) {4,8,12,16}
(ix) {2,6,10,14} (x) {5,10,15} (xi) {2,4,6,8,12,14,16} (xii) {5,15,20}
X = {a,b,c,d}, Y = {f,b,d,g}. Find: (i) X−Y (ii) Y−X (iii) X ∩ Y
Answer
(i) {a,c} (ii) {f,g} (iii) {b,d}
R is real numbers, Q is rationals. What is R − Q?
Answer
R − Q is the set of all irrational numbers.
True or false?
(i) {2,3,4,5} and {3,6} are disjoint (ii) {a,e,i,o,u} and {a,b,c,d} are disjoint
(iii) {2,6,10,14} and {3,7,11,15} are disjoint (iv) {2,6,10} and {3,7,11} are disjoint
Answer
(i) False (3 is common) (ii) False (a is common) (iii) True (iv) True
Exercise 1.5
U = {1,2,3,4,5,6,7,8,9}, A = {1,2,3,4}, B = {2,4,6,8}, C = {3,4,5,6}. Find:
Answer
(i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′ (vi) (B−C)′
Ans.
(i) A′ = {5,6,7,8,9}
(ii) B′ = {1,3,5,7,9}
(iii) A ∪ C = {1,2,3,4,5,6}; (A ∪ C)′ = {7,8,9}
(iv) A ∪ B = {1,2,3,4,6,8}; (A ∪ B)′ = {5,7,9}
(v) (A′)′ = A = {1,2,3,4}
(vi) B − C = {2,8}; (B−C)′ = {1,3,4,5,6,7,9}
U = {a,b,c,d,e,f,g,h}. Find complements:
Answer
(i) A = {a,b,c} (ii) B = {d,e,f,g} (iii) C = {a,c,e,g} (iv) D = {f,g,h,a}
Ans. (i) {d,e,f,g,h} (ii) {a,b,c,h} (iii) {b,d,f,h} (iv) {b,c,d,e}
Universal set = natural numbers N. Find complements:
(i) Even naturals (ii) Odd naturals (iii) Positive multiples of 3 (iv) Primes
(v) Naturals divisible by 3 and 5 (vi) Perfect squares (vii) Perfect cubes
Answer
(viii) {x : x + 5 = 8} (ix) {x : 2x + 5 = 9} (x) {x : x ≥ 7} (xi) {x : 2x + 1 > 10}
Ans.
(i) Odd naturals
(ii) Even naturals
(iii) {x ∈ N : x is not a multiple of 3}
(iv) {1} ∪ {composite numbers} = {x ∈ N : x is not prime}
(v) {x ∈ N : x is not divisible by 15}
(vi) {x ∈ N : x is not a perfect square}
(vii) {x ∈ N : x is not a perfect cube}
(viii) {3}′ = {x ∈ N : x ≠ 3}
(ix) {2}′ = {x ∈ N : x ≠ 2}
(x) {x ∈ N : x < 7} = {1,2,3,4,5,6}
(xi) 2x+1 > 10
⇒ x > 4.5, so set = {5,6,7,…}; complement = {1,2,3,4}
U = {1,2,3,4,5,6,7,8,9}, A = {2,4,6,8}, B = {2,3,5,7}. Verify:
Answer
(i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′
Ans.
A′ = {1,3,5,7,9}, B′ = {1,4,6,8,9}
(i) A ∪ B = {2,3,4,5,6,7,8}; (A ∪ B)′ = {1,9}. A′ ∩ B′ = {1,9}. Equal. ✓
(ii) A ∩ B = {2}; (A ∩ B)′ = {1,3,4,5,6,7,8,9}. A′ ∪ B′ = {1,3,4,5,6,7,8,9}. Equal. ✓
Draw Venn diagrams for: (i) (A ∪ B)′ (ii) A′ ∩ B′ (iii) (A ∩ B)′ (iv) A′ ∪ B′
Answer
(i) and (ii) are equal by De Morgan's law: shade the region outside both circles A and B.
(iii) and (iv) are equal by De Morgan's law: shade everything except the overlapping region of A and B.
U = all triangles in a plane. A = triangles with at least one angle ≠ 60°. What is A′?
Answer
A′ = set of equilateral triangles (all three angles exactly 60°).
Q7. Fill in the blanks
(i) A ∪ A′ = ___ (ii) φ′ ∩ A = ___ (iii) A ∩ A′ = ___ (iv) U′ ∩ A = ___
Ans. (i) U (ii) A (since φ′ = U) (iii) φ (iv) φ (since U′ = φ)
Miscellaneous Exercise on Chapter 1
Among A = {x ∈ R : x² − 8x + 12 = 0}, B = {2,4,6}, C = {2,4,6,8,…}, D = {6}, decide which sets are subsets of others.
Answer
x² − 8x + 12 = 0
⇒ (x−2)(x−6) = 0
⇒ A = {2, 6}.
D = {6} ⊂ A ⊂ B ⊂ C, and also D ⊂ B, D ⊂ C, A ⊂ C.
True or false? Prove if true, give counterexample if false.
(i) If x ∈ A and A ∈ B, then x ∈ B (ii) If A ⊂ B and B ∈ C, then A ∈ C
(iii) If A ⊂ B and B ⊂ C, then A ⊂ C (iv) If A ⊄ B and B ⊄ C, then A ⊄ C
(v) If x ∈ A and A ⊄ B, then x ∉ B (vi) If A ⊂ B and x ∉ B, then x ∉ A
Answer
(i) False. Let A = {1}, B = {{1}, 2}. Then 1 ∈ A, A ∈ B, but 1 ∉ B.
(ii) False. Let A = {1}, B = {1,2}, C = {{1,2}, 3}. A ⊂ B, B ∈ C, but A ∉ C.
(iii) True. If x ∈ A then x ∈ B (since A ⊂ B); then x ∈ C (since B ⊂ C). So A ⊂ C.
(iv) False. Let A = {1}, B = {2}, C = {1,3}. A ⊄ B and B ⊄ C, yet A ⊂ C.
(v) False. Let A = {1,2}, B = {2,3}. 1 ∈ A, A ⊄ B (since 1 ∉ B), yet 2 ∈ A and 2 ∈ B.
(vi) True. Suppose A ⊂ B and x ∉ B. If x ∈ A, then x ∈ B (since A ⊂ B) — contradiction. So x ∉ A.
A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show B = C.
Answer
Let b ∈ B. Then b ∈ A ∪ B = A ∪ C, so b ∈ A or b ∈ C. If b ∈ A: then b ∈ A ∩ B = A ∩ C, so b ∈ C. If b ∉ A: then since b ∈ A ∪ C, b ∈ C. In either case b ∈ C, so B ⊂ C. By symmetry C ⊂ B. Hence B = C.
Show four conditions are equivalent: (i) A ⊂ B, (ii) A − B = φ, (iii) A ∪ B = B, (iv) A ∩ B = A.
Answer
(i)⟹(ii): A ⊂ B means every element of A is in B, so nothing is in A but not B; A − B = φ.
(ii)⟹(iii): A − B = φ means every element of A is in B; so A ∪ B = B.
(iii)⟹(iv): A ∪ B = B. For any x ∈ A: x ∈ B (since A ⊂ A ∪ B = B), so x ∈ A ∩ B. Thus A ⊂ A ∩ B; and A ∩ B ⊂ A always; so A ∩ B = A.
(iv)⟹(i): A ∩ B = A means every element of A is in A ∩ B ⊂ B. So A ⊂ B.
Show that A ⊂ B implies C − B ⊂ C − A.
Answer
Let x ∈ C − B: x ∈ C and x ∉ B. If x ∈ A then x ∈ B (since A ⊂ B) — contradiction. So x ∉ A. Hence x ∈ C and x ∉ A, i.e., x ∈ C − A. Thus C − B ⊂ C − A.
Show A = (A ∩ B) ∪ (A − B) and A ∪ (B − A) = A ∪ B.
Answer
Part 1: Any x ∈ A either belongs to B (giving x ∈ A ∩ B) or does not (giving x ∈ A − B). So A ⊂ (A ∩ B) ∪ (A − B). Conversely both A ∩ B and A − B are subsets of A, so their union ⊂ A. Hence equality.
Part 2: Let x ∈ A ∪ (B − A). If x ∈ A: x ∈ A ∪ B. If x ∈ B − A: x ∈ B ⊂ A ∪ B. So A ∪ (B − A) ⊂ A ∪ B. Conversely, if x ∈ A ∪ B: if x ∈ A, done; if x ∉ A but x ∈ B, then x ∈ B − A ⊂ A ∪ (B − A). Hence equality.
Using set properties, show: (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A
Answer
(i) A ∩ B ⊂ A always, so A ∪ (A ∩ B) = A (absorption law).
(ii) A ⊂ A ∪ B always, so A ∩ (A ∪ B) = A (absorption law).
Show A ∩ B = A ∩ C need not imply B = C.
Answer
Let A = {1,2}, B = {2,3}, C = {2,4}. Then A ∩ B = {2} = A ∩ C, but B ≠ C.
Given A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X, show A = B.
Answer
A = A ∩ (A ∪ X) = A ∩ (B ∪ X) = (A ∩ B) ∪ (A ∩ X) = (A ∩ B) ∪ φ = A ∩ B. So A ⊂ B. Similarly B = B ∩ (B ∪ X) = B ∩ (A ∪ X) = (B ∩ A) ∪ (B ∩ X) = A ∩ B. So B ⊂ A. Hence A = B.
Find sets A, B, C such that A ∩ B, B ∩ C, A ∩ C are non-empty but A ∩ B ∩ C = φ.
Answer
A = {1,2}, B = {2,3}, C = {1,3}. Then A ∩ B = {2}, B ∩ C = {3}, A ∩ C = {1}, all non-empty; but A ∩ B ∩ C = φ.
Important Extra Questions
If A = {1,2,3,4,5}, how many subsets and proper subsets does A have?
Answer
Subsets = 25 = 32. Proper subsets = 32 − 1 = 31.
If n(A) = 20, n(B) = 25, n(A ∪ B) = 40, find n(A ∩ B).
Answer
n(A ∩ B) = n(A) + n(B) − n(A ∪ B) = 20 + 25 − 40 = 5.
State De Morgan's laws for sets.
Answer
For any sets A and B in a universal set U:
(i) (A ∪ B)′ = A′ ∩ B′
(ii) (A ∩ B)′ = A′ ∪ B′
Show A ∪ φ = A and A ∩ U = A.
Answer
φ has no elements, so A ∪ φ contains only elements of A: A ∪ φ = A. Every element of A belongs to the universal set U, so A ∩ U = A.
In a class of 50 students, 30 play cricket, 25 play football, 15 play both. How many play neither?
Answer
n(C ∪ F) = 30 + 25 − 15 = 40. Neither = 50 − 40 = 10.