RS AGGARWAL CLASS 11 MATHS SOLUTION SET THEORY-2

RS AGGARWAL CLASS 11 MATHS SOLUTION SET THEORY-2

RS AGGARWAL CLASS 11 MATHS FREE SOLUTION SET THEORY (EXERCISE-1D, 1E, 1F)

R. S. AGGARWAL
CHAPTER 1 -SETS

Exercise 1D

Q. 1. (i) If A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n}, find: A ∪ B

Answer : Given: A = {a, b, c, d, e, f} and
B = {c, e, g, h} 
∴ A ∪ B = {a, b, c, d, e, f, g, h}

Q. 1. (ii)  If A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n}, find: B ∪ C

Answer : Given: B = {c, e, g, h} and
C = {7, 8, 9, 10, 11}
∴ B ∪ C = {a, c, e, g, h, m, n}

Q. 1. (iii) If A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n}, find: A ∪ C

Answer : Given; A = {a, b, c, d, e, f} and
C = {7, 8, 9, 10, 11}
∴ A ∪ C = {a, b, c, d, e, f, m, n}

Q. 1. (iv) If A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n}, find: B ∩ C

Answer : Given; B = {c, e, g, h} and
C = {7, 8, 9, 10, 11}
∴ B ∩ C ={e}

Q. 1. (v) If A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n}, find: C ∩ A

Answer : Given; A = {a, b, c, d, e, f} and
C = {7, 8, 9, 10, 11}
∴ C ∩ A = {a, e}

Q. 1. (vi) If A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n}, find: A ∩ B

Answer : Given; A = {a, b, c, d, e, f} and
B = {c, e, g, h} A ∩ B = {c, e}

Q. 2. (i) If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find: A ∪ B

Answer: Given A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}
∴ A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}

Q. 2. (ii) If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find:  B ∪ C

Answer: Given B = {4, 5, 6, 7, 8} and
C = {7, 8, 9, 10, 11}
∴ B ∪ C = {4, 5, 6, 7, 8, 9, 10, 11}

Q. 2. (iii) If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find: A ∪ C

Answer: Given A = {1, 2, 3, 4, 5} and
C = {7, 8, 9, 10, 11}
∴ A ∪ C = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}

Exercise 1D

Q. 2. (iv) If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find: B ∪ D

Answer: Given B = {4, 5, 6, 7, 8} and
D = {10, 11, 12, 13, 14}
∴ B ∪ D = {4, 5, 6, 7, 8, 10, 11, 12, 13, 14}.

Q. 2. (v) If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find: (A ∪ B) ∪ C

Answer: Given A = {1, 2, 3, 4, 5},
B = {4, 5, 6, 7, 8} and
C = {7, 8, 9, 10, 11},
∴ (A ∪ B) ∪ C
= {1, 2, 3, 4, 5, 6, 7, 8} ∪ {7, 8, 9, 10, 11}
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Q. 2. (vi) If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find: (A ∪ B) ∩ C

Answer: Given A = {1, 2, 3, 4, 5},
B = {4, 5, 6, 7, 8} and
C = {7, 8, 9, 10, 11}
∴ (A ∪ B) ∩ C
= {1, 2, 3, 4, 5, 6, 7, 8} ∩ {7, 8, 9, 10, 11}
= {7, 8}

Exercise 1D SETS

Q. 2. (vii) If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find: (A ∩ B) ∪ D

Answer: Given A = {1, 2, 3, 4, 5},
B = {4, 5, 6, 7, 8} and
D = {10, 11, 12, 13, 14}
∴ (A ∩ B) ∪ D
= {4, 5} ∪ {10, 11, 12, 13, 14}
= {4, 5, 10, 11, 12, 13, 14}

Q. 2. (viii) If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find: (A ∩ B) ∪ (B ∩ C)

Answer: Given A = {1, 2, 3, 4, 5},
B = {4, 5, 6, 7, 8} and
C = {7, 8, 9, 10, 11}
∴ (A ∩ B) ∪ (B ∩ C)
= {4, 5} ∪ {7, 8}
= {4, 5, 7, 8}

Q. 2. (ix) If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}, find: (A ∪ C) ∩ (C ∪ D)

Answer: Given A = {1, 2, 3, 4, 5},
C = {7, 8, 9, 10, 11} and
D = {10, 11, 12, 13, 14}
∴ (A ∪ C) ∩ (C ∪ D}
= {1, 2, 3, 4, 5, 7, 8, 9, 10, 11} ∩ {7, 8, 9, 10, 11, 12, 13, 14}
= {7, 8, 9, 10, 11} 

Q. 3. (i) If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}, and D = {15, 17}, find: A ∩ B

Answer: Given: A = {3, 5, 7, 9, 11} and
B = {7, 9, 11, 13}
∴ A ∩ B = {7, 9, 11}

Q. 3. (ii) If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}, and D = {15, 17}, find: A ∩ C

Answer: Given: A = {3, 5, 7, 9, 11} and
C = {11, 13, 15}
∴ A ∩ C = {11}

Exercise 1D

Q. 3. (iii) If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}, and D = {15, 17}, find: B ∩ C
Answer: Given: B = {7, 9, 11, 13} and
C = {11, 13, 15}
∴ B ∩ C = {11, 13}

Q. 3. (iv) If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}, and D = {15, 17}, find: B ∩ D
Answer: Given: B = {7, 9, 11, 13} and
D = {15, 17}
∴ B ∩ D = Φ

Q. 3. (v) If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}, and D = {15, 17}, find: B ∩ (C ∪ D)
Answer: Given: B = {7, 9, 11, 13},
C = {11, 13, 15} and
D = {15, 17}
∴ B ∩ (C ∪ D)
= {7, 9, 11, 13} ∩  11, 13, 15, 17}
=  {11, 13}

Q. 3. (vi) If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}, and D = {15, 17}, find: A ∩ (B ∪ C)
Answer: Given: A = {3, 5, 7, 9, 11}
B = {7, 9, 11, 13} and
C = {11, 13, 15}
∴ A ∩ (B ∪ C)
=  {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15}
=  {7, 9, 11}

Q. 4. (i) If A = {x : x ∈ N}, B = {x : x ∈ N and x is even), C = {x : x ∈ N and x is odd} and D = {x : x ∈ N and x is prime} then find: A ∩ B
Answer: Given: A = {x : x ∈ N} and
B = {x : x ∈ N and x is even)
∴ A ∩ B = {x : x ∈ N and x is even} = B

Q. 4. (ii) If A = {x : x ∈ N}, B = {x : x ∈ N and x is even), C = {x : x ∈ N and x is odd} and D = {x : x ∈ N and x is prime} then find: A ∩ C
Answer: Given: A = {x : x ∈ N} and
C = {x : x ∈ N and x is odd}
∴ A ∩ C = {x : x ∈ N and x is odd} = C

Q. 4. (iii) If A = {x : x ∈ N}, B = {x : x ∈ N and x is even), C = {x : x ∈ N and x is odd} and D = {x : x ∈ N and x is prime} then find: A ∩ D
Answer: Given: A = {x : x ∈ N} and
D = {x : x ∈ N and x is prime}
∴ A ∩ D = {x : x ∈ N and x is prime} = D

Q. 4. (iv) If A = {x : x ∈ N}, B = {x : x ∈ N and x is even), C = {x : x ∈ N and x is odd} and D = {x : x ∈ N and x is prime} then find: B ∩ C
Answer: Given: B = {x : x ∈ N and x is even) and
C = {x : x ∈ N and x is odd}
∴ B ∩ C = Φ

Q. 4. (v) If A = {x : x ∈ N}, B = {x : x ∈ N and x is even), C = {x : x ∈ N and x is odd} and D = {x : x ∈ N and x is prime} then find: B ∩ D
Answer: Given: B = {x : x ∈ N and x is even) and
D = {x : x ∈ N and x is prime}
∴ B ∩ D = {2}
[∵ 2 is the only even prime number]

Q. 4. (vi) If A = {x : x ∈ N}, B = {x : x ∈ N and x is even), C = {x : x ∈ N and x is odd} and D = {x : x ∈ N and x is prime} then find:  C ∩ D
Answer: Given: C = {x : x ∈ N and x is odd} and
D = {x : x ∈ N and x is prime}
∴ C ∩ D
= {x : x ∈ N and x is prime and x ≠ 2}
= D – {2}

Exercise 1D

Q. 5. (i) If A = {2x : x ∈ N}, 1 ≤ x < 4}, B = {x + 2) : x ∈ N and 2 ≤ x < 5} and C = {x : x ∈ N and 4 < x < 8}, find: A ∩ B
Answer: Given: A = {2x : x ∈ N}, 1 ≤ x < 4} and
B = {x + 2) : x ∈ N and 2 ≤ x < 5}
∴ A = {2, 4, 6},
B = {4, 5, 6}
∴  A ∩ B = {4, 6}

Q. 5. (ii) If A = {2x : x ∈ N}, 1 ≤ x < 4}, B = {x + 2) : x ∈ N and 2 ≤ x < 5} and C = {x : x ∈ N and 4 < x < 8}, find: A ∪ B
Answer: Given: A = {2x : x ∈ N}1 ≤ x < 4} and
B = {x + 2) : x ∈ N and 2 ≤ x < 5}
∴ A = {2, 4, 6},
B = {4, 5, 6}
∴ A ∪ B = {2, 4, 5, 6}

Q. 5. (iii) If A = {2x : x ∈ N}, 1 ≤ x < 4}, B = {x + 2) : x ∈ N and 2 ≤ x < 5} and C = {x : x ∈ N and 4 < x < 8}, find: (A ∪ B) ∩ C
Answer: Given: A = {2x : x ∈ N}, 1 ≤ x < 4},
B = {x + 2) : x ∈ N and 2 ≤ x < 5} and
C = {x : x ∈ N and 4 < x < 8}
∴ A = {2, 4, 6},
B = {4, 5, 6} and
C = {5, 6, 7}
∴ (A ∪ B) ∩ C
= {2, 4, 5, 6} ∩ {5, 6, 7}
= {5, 6}

Q. 6. (i) If A = {2, 4, 6, 8, 10, 12}, B = {3, 4, 5, 6, 7, 8, 10}, find: (A – B)
Answer: Given: A = {2, 4, 6, 8, 10, 12},
B = {3, 4, 5, 6, 7, 8, 10}
∴ (A – B) = {2, 12}

Q. 6. (ii) If A = {2, 4, 6, 8, 10, 12}, B = {3, 4, 5, 6, 7, 8, 10}, find: (B – A)
Answer: Given: A = {2, 4, 6, 8, 10, 12},
B = {3, 4, 5, 6, 7, 8, 10}
∴ (B – A) = {3, 5, 7}

Q. 6. (iii) If A = {2, 4, 6, 8, 10, 12}, B = {3, 4, 5, 6, 7, 8, 10}, find: (A – B) ∪ (B – A)
Answer: Given: A = {2, 4, 6, 8, 10, 12},
B = {3, 4, 5, 6, 7, 8, 10}
∴ (A – B) ∪ (B – A)
= {2, 12} ∪ {3, 5, 7}
= {2, 3, 5, 7, 12}

Q. 7. (i) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}, find: A ∩ (B – C)
Answer: Given: A = {a, b, c, d, e},
B = {a, c, e, g} and
C = {b, e, f, g}
∴ A ∩ (B – C)
⇒ {a, b, c, d, e} ∩ {a, c}
⇒ {a, c}

Q. 7. (ii) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}, find: A – (B ∪ C)
Answer: Given: A = {a, b, c, d, e},
B = {a, c, e, g} and
C = {b, e, f, g}
∴ A – (B ∪ C)
= {a, b, c, d, e} – {a, b, c, e, f, g}
= {d}

Q. 7. (iii) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}, find: A – (B ∩ C)

Answer: Given: A = {a, b, c, d, e},
B = {a, c, e, g} and
C = {b, e, f, g}
∴ A – (B ∩ C)
⇒ {a, b, c, d, e} – {e, g}
= {a, b, c, d} 

Q. 8. (i) If A = {¹/_x: x ∈ N and x < 8}, and B = {¹/_2x: x ∈ N and x ≤ 4} , find : A ∪ B
Answer: Given A = {¹/_x: x ∈ N and x < 8} and
B = {¹/_2x: x ∈ N and x ≤ 4}
∴ A = {1, ½, ⅓, ¼, ⅕, ⅙, ¹/₇}
B = {½, ¼, ⅙, ¹/₈}
∴  A ∪ B = {1, ½, ⅓, ¼, ⅕, ⅙, ¹/₇, ¹/₈}

Q. 8. (ii) If A = {¹/_x: x ∈ N and x < 8}, and B = {¹/_2x: x ∈ N and x ≤ 4} , find : A ∩ B
Answer: Given A = {¹/_x: x ∈ N and x < 8} and
B = {¹/_2x: x ∈ N and x ≤ 4}
∴ A = {1, ½, ⅓, ¼, ⅕, ⅙, ¹/₇}
B = {½, ¼, ⅙, ¹/₈}
∴ A ∩ B = {½, ¼, ⅙}

Q. 8. (iii) If A = {¹/_x: x ∈ N and x < 8}, and B = {¹/_2x: x ∈ N and x ≤ 4} , find : A – B
Answer: Given A = {¹/_x: x ∈ N and x < 8} and
B = {¹/_2x: x ∈ N and x ≤ 4}
∴ A = {1, ½, ⅓, ¼, ⅕, ⅙, ¹/₇}
B = {½, ¼, ⅙, ¹/₈}
∴ A – B =  {1, ⅓, ⅕, ¹/₇}Q. 8. (iv) If A = {¹/_x: x ∈ N and x < 8}, and B = {¹/_2x: x ∈ N and x ≤ 4} , find : B – A
Answer: Given A = {¹/_x: x ∈ N and x < 8} and
B = {¹/_2x: x ∈ N and x ≤ 4}
∴ A = {1, ½, ⅓, ¼, ⅕, ⅙, ¹/₇}
B = {½, ¼, ⅙, ¹/₈}
∴ B – A =  {¹/₈}

Q. 9. If R is the set of all real numbers and Q is the set of all rational numbers then what is the set (R – Q)? 
Answer: Given; R is the set of all real numbers and Q is the set of all rational numbers. 
Real numbers are the set of combinations of rational numbers and irrational numbers.
So (R – Q) is the set of all irrational numbers.
∴ R – Q = {x: x ∈ N and x is irrational numbers}

Q. 10. (i) If A = {2, 3, 5, 7, 11} and B = ϕ, find:  A ∪ B
Answer: Given: A = {2, 3, 5, 7, 11} and B = Φ
∴ A ∪ B = {2, 3, 5, 7, 11}

Q. 10. (ii) If A = {2, 3, 5, 7, 11} and B = ϕ, find: A ∩ B
Answer: Given: A = {2, 3, 5, 7, 11} and B = Φ
∴  A ∩ B = Φ 

Q. 11. (i) If A and B are two sets such that A ⊆ B then find: A ∪ B
Answer: ∵ A ⊆ B
Therefore B consists of all the elements of A.
∴ A ∪ B = B

Q. 11. (ii) If A and B are two sets such that A ⊆ B then find:  A ∩ B
Answer: A is a subset of B, so all the elements of A are contained in B.
∴ A ∩ B = A

Q. 11. (iii) If A and B are two sets such that A ⊆ B then find:  A – B
Answer: A – B
= A – (A ∩ B)
= A – A …… [∵ A ⊆ B]
= Φ

Q. 12. (i) Which of the following sets are pairs of disjoint sets? Justify your answer. A = {3, 4, 5, 6} and B = {2, 5, 7, 9}
[Two sets are said to be disjoint sets if they have no element in common i.e. their intersection is the empty set.]
Answer:   Given A = {3, 4, 5, 6} and
B = {2, 5, 7, 9}
∴ A ∩ B = {5}
 ∴ A and B are not pairs of disjoint sets.

Q. 12. (ii) Which of the following sets are pairs of disjoint sets? Justify your answer. C = {1, 2, 3, 4, 5} and D = {6, 7, 9, 11}
Answer:  C = {1, 2, 3, 4, 5} and
D = {6, 7, 9, 11}
∴ C ∩ D = Φ
∴ C and D are pairs of disjoint sets.

Q. 12. (iii) Which of the following sets are pairs of disjoint sets? Justify your answer.  E = {x : x ∈ N, x is even and x < 8} F = {x : x = 3n, n ∈ N, and x < 4}
Answer: E = {x : x ∈ N, x is even and x < 8}
= {2, 4, 6} and
F = {x : x = 3n, n ∈ N, and x < 4}
= {3, 6, 9} ∴ E ∩ F = {6}
∴ E and F are not pairs of disjoint sets.

Q. 12. (iv) Which of the following sets are pairs of disjoint sets? Justify your answer. G = {x : x ∈ N, x is even} and H {x : x ∈ N, x is prime}
Answer: G = {x : x ∈ N, x is even} and
H {x : x ∈ N, x is prime}
∵ Only 2 is an even prime number;
∴ G ∩ G = {2}
∴ G and H are not pairs of disjoint sets.

Q. 12. (v) Which of the following sets are pairs of disjoint sets? Justify your answer.  J = {x : x ∈ N, x is even} and K = {x : x ∈ N, x is odd}
Answer: J = {x : x ∈ N, x is even} and
K = {x : x ∈ N, x is odd}
∵ there is not any number which is both odd and even simultaneously.
∴ J and K are pairs of disjoint sets.

Q. 13. (i) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find:  A’
Answer: Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and
A = {1, 2, 3, 4}
∴ A’ = {5, 6, 7, 8, 9}

Q. 13. (ii) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find: B’
Answer: Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and
B = {2, 4, 6, 8}
∴ B’ = {1, 3, 5, 7, 9}

Q. 13. (iii) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find: C’
Answer: Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and
C = {1, 4, 5, 6}
∴ C’ = {2, 3, 7, 8, 9}

Q. 13. (iv) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find: (B’)’
Answer: Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and
B = {2, 4, 6, 8}
∴ (B’)’ = B = {2, 4, 6, 8}

Q. 13. (v) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find:  (A ∪ B)’
Answer: Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
A = {1, 2, 3, 4} and
B = {2, 4, 6, 8}
∴ (A ∪ B) =  {1, 2, 3, 4, 6, 8}
∴ (A ∪ B)’ = {5, 7, 9}

Q. 13. (vi) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find: (A ∩ C)’
Answer: Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
A = {1, 2, 3, 4} and
C = {1, 4, 5, 6}
∴ A ∩ C = {1, 4}
∴ (A ∩ C)’ = {2, 3, 5, 6, 7, 8, 9}

Q. 13. (vii) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find: ( B – C)’
Answer: Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
A = {1, 2, 3, 4} and C = {1, 4, 5, 6} 
∴ A ∩ C = {1, 4}
B – C = {2, 8}
∴ (B – C)’
= {1, 3, 4, 5, 6, 7, 9}

Q. 14. (i) if U = {a, b, c, d, e},  A = {a, b, c} and B = {b, c, d, e} then verify that: (A ∪ B)’ = (A’ ∩ B’)

Answer: Given: U = {a, b, c, d, e},
A = {a, b, c}  and
B = {a, c, d, e}
∴ A ∪ B =  {a, b, c, d, e}
L.H.S.
= (A ∪ B)’ = Φ
A’ =  {d, e}
B’ =  {b}
R.H.S.
(A’ ∩ B’) = Φ = L.H.S.
∴ (A ∪ B)’ = (A’ ∩ B’) (Proved)

Q. 14. (ii) if U = {a, b, c, d, e},  A = {a, b, c} and B = {b, c, d, e} then verify that: (A ∩ B)’ = (A’ ∪ B’)

Answer: Given: U = {a, b, c, d, e},
A = {a, b, c}  and
B = {a, c, d, e}
∴ A ∩ B = {a, c}
L.H.S.
(A ∩ B)’ = {b, d, e}
A’ =  {d, e}
B’ =  {b}
R.H.S.
A’ ∪ B’ = {b, d, e} = L.H.S.
∴  (A ∩ B)’ = (A’ ∪ B’) (Proved)

Q.15. if U is the universal set and A ⊂ U then fill in the blanks. 
(i) A ∪ A’ = …. 
(ii) A ∩ A’ = …. 
(iii) Φ‘ ∩ A = …. 
(iv) U’ ∩ A = …. 

Answer: Given; U is the universal set and A ⊂ U 
(i) A ∪ A’ = U 
(ii) A ∩ A’ = Φ 
(iii) Φ‘ ∩ A = A 
(iv) U’ ∩ A = Φ 

Exercise 1E

Q. 1. (i) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g} verify that: (i) A ∪ B = B ∪ A

Answer: L.H.S.
= A ∪ B
⇒ {a, b, c, d, e} ∪ {a, c, e, g}
= {a, b, c, d, e, g}
R.H.S.
= B ∪ A
= {a, c, e, g} ∪ {a, b, c, d, e}
⇒ {a, b, c, d, e, g} = L.H.S.
∴ A ∪ B = B ∪ A (Proved)

Q. 1. (ii) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g} verify that: A ∪ C = C ∪ A

Answer: L.H.S.
= A ∪ C
⇒ {a, b, c, d, e} ∪ {b, e, f, g}
= {a, b, c, d, e, f, g}
R.H.S.
= C ∪ A
⇒ {b, e, f, g} ∪ {a, b, c, d, e}
= {a, b, c, d, e, f, g}= L.H.S.
∴ A ∪ C = C ∪ A   (Proved)

Q. 1. (iii) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g} verify that: B ∪ C = C ∪ B

Answer: L.H.S.
= B ∪ C
⇒ {a, c, e, g} ∪ {b, e, f, g}
= {a, b, c, e, f, g}
R.H.S.
= C ∪ B
⇒ {b, e, f, g} ∪ {a, c, e, g}
= {a, b, c, d, e, f, g} = L.H.S.
∴ B ∪ C = C ∪ B   (Proved)

Exercise 1E

Q. 1. (iv) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g} verify that: A ∩ B = B ∩ A

Answer: L.H.S.
= A ∩ B
⇒ {a, b, c, d, e} ∩ {a, c, e, g}
= {a, c, e}
R.H.S.
= B ∩ A
= {a, c, e, g} ∪ {a, b, c, d, e}
= { a, c, e} = L.H.S.
∴ A ∩ B = B ∩ A  (Proved)

Q. 1. (v) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g} verify that:
B ∩ C = C ∩ B

Answer: L.H.S.
= B ∩ C
⇒ {a, c, e, g} ∩ {b, e, f, g}
= {e, g}
R.H.S.
= C ∩ B
⇒ {b, e, f, g} ∩ {a, c, e, g}
= {e, g} = L.H.S.
∴ B ∩ C = C ∩ B  (Proved)

Q. 1. (vi) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g} verify that: A ∩ C = C ∩ A

Answer: L.H.S.
= A ∩ C
= {a, b, c, d, e} ∩ {b, e, f, g}
= {b, e}
R.H.S.
= C ∩ A
= {b, e, f, g} ∪ {a, b, c, d, e}
= {b, e} = L.H.S.
∴ A ∩ C = C ∩ A  (Proved)

Q. 1. (vii) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g} verify that: (A ∪ B) ∪ C = A ∪ (B ∪ C)

Answer: L.H.S.
= (A ∪ B) ∪ C
= {a, b, c, d, e, g} ∪ {b, e, f, g}
= {a, b, c, d, e, f, g}
R.H.S.
= A ∪ (B ∪ C)
= {a, b, c, d, e} ∪ {a, b, c, d, e, f, g}
= {a, b, c, d, e, f, g} = L.H.S.
∴ (A ∪ B) ∪ C = A ∪ (B ∪ C)  (Proved)

Q. 1. (viii) If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g} verify that: (A ∩ B) ∩ C = A ∩ (B ∩ C)

Answer: L.H.S.
= (A ∩ B) ∩ C
= {a, c, e} ∩ {b, e, f, g}
= {e}
R.H.S.
= A ∩ (B ∩ C)
= {a, b, c, d, e} ∩ {e, g}
= {e} = L.H.S.
∴ (A ∩ B) ∩ C = A ∩ (B ∩ C)  (Proved) 

Q. 2. (i) If A = {a, b, c, d, e}, B = {a, c, e, g}, and C = {b, e, f, g} verify that: A ∩ (B – C) = (A ∩ B) – (A ∩ C)

Answer: A = {a, b, c, d, e},
B = {a, c, e, g}, and
C = {b, e, f, g}
L.H.S.
= A ∩ (B – C)
= {a, b, c, d, e} ∩ {a, c}
= {a, c}
R.H.S.
= (A ∩ B) – (A ∩ C)
= {a, c, e} ∩ {b, e}
= {a, c} = L.H.S.
∴ A ∩ (B – C) = (A ∩ B) – (A ∩ C)   (Proved) 

Q. 2. (ii) If A = {a, b, c, d, e}, B = {a, c, e, g}, and C = {b, e, f, g} verify that: (ii) A – (B ∩ C) = (A – B) ∪ (A – C)

Answer: L.H.S.
= A – (B ∩ C)
= {a, b, c, d, e} – {e, g}
= {a, b, c, d}
R.H.S.
= (A – B) ∪ (A – C)
= {b, d} ∪ {a, c, d}
= {a, b, c, d} = L.H.S.
∴ A – (B ∩ C) = (A – B) ∪ (A – C)   (Proved) 

Q. 3. (i) If A = {x : x ε N, x ≤ 7}, B = {x : x is prime, x < 8} and C = {x : x ε N, x is odd and x < 10}, verify that: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Answer: A = {x : x ε N, x ≤ 7}
⇒ A = {1, 2, 3, 4, 5, 6, 7}
B = {x : x is prime, x < 8}
⇒ B = {2, 3, 5, 7}
C = {x : x ε N, x is odd and x < 10}
⇒ C = {1, 3, 5, 7, 9}
B ∩ C = {3, 5, 7}
L.H.S.
= A U (B ∩ C)
= {1, 2, 3, 4, 5, 6, 7} A ∪ B
= {1, 2, 3, 4, 5, 6, 7} A ∪ C
= {1, 2, 3, 4, 5, 6, 7, 9}
R.H.S.
= (A ∪ B) ∩ (A ∪ C)
= {1, 2, 3, 4, 5, 6, 7} = L.H.S.
∴ A U (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (Proved)

Q. 3. (ii) If A = {x : x ε N, x ≤ 7}, B = {x : x is prime, x < 8} and C = {x : x ε N, x is odd and x < 10}, verify that:  A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Answer : A = {x : x ε N, x ≤ 7}
⇒ A = {1, 2, 3, 4, 5, 6, 7}
B = {x : x is prime, x < 8}
⇒ B = {2, 3, 5, 7}
C = {x : x ε N, x is odd and x < 10}
⇒ C = {1, 3, 5, 7, 9}
B ∪ C = {1, 2, 3, 5, 7, 9}
L.H.S.
= A ∩ (B ∪ C)
= {1, 2, 3, 5, 7} A ∩ B
= {2, 3, 5, 7} A ∩ C
= {1, 3, 5, 7}
R.H.S.
= (A ∩ B) ∪ (A ∩ C)
= {1, 2, 3, 5, 7}= L.H.S.
∴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)  (Proved) 

Q. 4. (i) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, and B = {2, 3, 5, 7} verify that: (A ∪ B)’ = (A’ ∩ B’)

Answer: A ∪ B = {2, 3, 4, 5, 6, 7, 8}
L.H.S.
= (A ∩ B)’ = {1, 9}
A’ = {1, 3, 5, 7, 9}
B’ = {1, 4, 6, 8, 9}
R.H.S.
= A’ ∩ B’ = {1, 9} = L.H.S.
∴ (A ∪ B)’ = (A’ ∩ B’)  (Proved)

Q. 4. (ii) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, and B = {2, 3, 5, 7} verify that:  (A ∩ B)’ = (A’ ∪ B’)

Answer:  A ∩ B = {2}
L.H.S.
= (A ∩ B)’
= {1, 3, 4, 5, 6, 7, 8, 9}
A’ = {1, 3, 5, 7}
B’ = {1, 4, 6, 8, 9}
R.H.S.
= A’ ∪ B’
= {1, 3, 4, 5, 6, 7, 8, 9} = L.H.S.
∴  (A ∩ C)’ = A’ U B’  (Proved) 

Q. 5. (i) Let A = {a, b, c}, B = {b, c, d, e} and C = {c, d, e, f} be subsets of U = {a, b, c, d, e, f}. Then verify that: (A’)’ = A

Answer: A’
= U – A
= {a, b, c, d, e, f} – {a, b, c}
={d, e, f}
(A’)’ = U – A’
= {a, b, c, d, e, f} – {d, e, f}
= {a, b, c} = A (Proved)

Q. 5. (ii) Let A = {a, b, c}, B = {b, c, d, e} and C = {c, d, e, f} be subsets of U = {a, b, c, d, e, f}. Then verify that:  (A ∪ B)’ = (A’ ∩ B’)

Answer: A ∪ B = {a, b, c, d, e}
L.H.S.
= (A ∪ B)’
= U – (A ∪ B)
= {a, b, c, d, e, f} – {a, b, c, d, e}
= {f}
A’
= U – A
= {a, b, c, d, e, f} – {a, b, c}
= {d, e, f}
B’
= U – B
= {a, b, c, d, e, f} – {b, c, d, e,}
{a, f}
R.H.S.
= A’ ∩ B’ = {f} = L.H.S.
∴ (A ∪ B)’ = (A’ ∩ B’) (Proved)

Q. 5. (iii) Let A = {a, b, c}, B = {b, c, d, e} and C = {c, d, e, f} be subsets of U = {a, b, c, d, e, f}. Then verify that:  (A ∩ B)’ = (A’ ∪ B’)

Answer: A ∩ B = {b, c}
L.H.S.
= (A ∩ B)’
= U – (A ∩ B)
= {a, b, c, d, e, f} – {b, c}
{a, d, e, f}
A’
= U – A
= {a, b, c, d, e, f} – {a, b, c}
= {d, e, f}
B’
= U – B
= {a, b, c, d, e, f} – {b, c, d, e,}
{a, f}
R.H.S.
= A’ ∪ B’ = {a, d, e, f} = L.H.S.
∴ (A ∩ B)’ = (A’ ∪ B’)  (Proved) 

Q. 6. Given an example of three sets A, B, C such that A ∩ B ≠ φ, B ∩ C ≠ φ, A ∩ C ≠ φ, and A ∩ B ∩ C = φ

Answer: Let A = {a, b}; B = {b, c } and C = {a, c}
∴ A ∩ B = {b} ≠ φ,
∴ B ∩ C = {c} ≠ φ,
and A ∩ C = {a} ≠ φ,
∴ A ∩ B ∩ C = {} = φ
Ans: Three sets are {a, b}, {b, c }, {a, c} and satisfy the given conditions.

Q. 7. (i) For any sets A and B, prove that: (i) (A – B) ∩ B = φ 

Answer: If possible,
let x ∈ (A – B) ∩ B
⇒ (x ∈ A and x ∉ B) and x ∈ B
Hence x ∉ B and x ∈ B
This contradicts the fact.
It is possible if there is no such element x.
Therefore (A – B) ∩ B is empty i.e. (A – B) ∩ B = φ (Proved)

Q. 7. (ii) For any sets A and B, prove that: A ∪ (B – A) = A ∪ B

Answer: L.H.S.
= A ∪ (B – A)
= A ∪ (B ∩ A^c)
⇒ (A ∪ B) ∩ (A ∪ A^c)… [Distribution Law]
= (A ∪ B) ∩ S
= (A ∪ B) =  R.H.S.  (Proved)

Q. 7. (iii) For any sets A and B, prove that: (A – B) ∪ (A ∩ B) = A – B

Answer: L.H.S.
=  (A – B) ∪ (A ∩ B)
= (A ∩ B^c) ∪ (A ∩ B)
⇒ [(A ∩ B^c) ∪ A] ∩ [(A ∩ B^c) ∪ B]… [Distribution Law]
= [(A ∪ A) ∩ (B^c ∪ A)] ∩ [(A ∪ B) ∩ (B^c ∪ B)]… [Distribution Law]
= [A ∩ (B^c ∪ A)] ∩ [(A ∪ B) ∩ S]… [S= Universal Set]
⇒ [(A ∩ B^c) ∪ (A ∩ A)] ∩ (A ∪ B)
= (A – B) ∩ (A ∪ B)
= (A – B) = R.H.S.  (Proved)

Q. 7. (iv) For any sets A and B, prove that: (A ∪ B) – B = A – B

Answer: L.H.S.
=  (A – B) ∪ (A ∩ B)
= (A ∩ B^c) ∪ (A ∩ B)
⇒ [(A ∩ B^c) ∪ A] ∩ [(A ∩ B^c) ∪ B]… [Distribution Law]
= [(A ∪ A) ∩ (B^c ∪ A)] ∩ [(A ∪ B) ∩ (B^c ∪ B)]… [Distribution Law]
= [A ∩ (B^c ∪ A)] ∩ [(A ∪ B) ∩ S]… [S= Universal Set]
⇒ [(A ∩ B^c) ∪ (A ∩ A)] ∩ (A ∪ B)
= (A – B) ∩ (A ∪ B)
= (A – B) = R.H.S. (Proved) 

Q. 7. (iv) For any sets A and B, prove that: (A ∪ B) – B = A – B

Answer: L.H.S.
= (A ∪ B) – B
=  (A ∪ B) ∩ B^c
⇒ (A ∪ B^c) ∩ (B ∪ B^c) … [Distribution Law]
= (A ∪ B^c) ∩ S … [S= Universal Set]
= A ∪ B^c = A – B = R.H.S.  (Proved)

Q. 7.(v) For any sets A and B, prove that: A – (A ∩ B) = A – B

Answer:  L.H.S.
= A – (A ∪ B)
= A ∩ (A ∪ B)^c
⇒ A ∩ (A^c ∩ B^c)….[De Morgan’s Law]
⇒ (A ∩ A^c) ∩ B^c
= A ∩ B^c
= A – B = R.H.S.  (Proved) 

Q. 8. (i) For any sets A and B, prove that:  A ∩ B’ = φ ⇒ A ⊂ B

Answer: Let x ∈ A
∴ x ∉ B’……. [∴ A ∩ B’ = φ]
Again x ∉ B’ ⇒ x ∈ B
Since x ∈ A ⇒ x ∈ B
∴ A ⊂ B  (Proved)

Q. 8. (ii) For any setschecking, prove that: A’ ∪ B = U ⇒ A ⊂ B

Answer: Let x ∈ A
If x ∈ A then x ∉ A’
Since A’ ∪ B = U and x ∉ A’,
Then x must be in B
∴ A ⊂ B  (Proved)

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Exercise 1F

1. (i) Let A = {a, b, c, e, f} B = {c, d, e, g} and C = {b, c, f, g} be subsets of the set U = {a, b, c, d, e, f, g, h}. Draw Venn diagrams to represent the following sets: A ∩ B

Answer: Given: A = {a, b, c, e, f},
B = {c, d, e, g} and
U = {a, b, c, d, e, f, g, h}
Venn diagrams:
Two sets A and B are shown with the following Venn Diagrams.
A set denoted by red circle and B set denoted by yellow circle.