Referring to the AP Inter 1st Year Maths Study Material Chapter 1 Sets Exercise 1d Solutions makes it easier to understand complex problems.
AP Inter 1st Year Maths Sets Solutions Exercise 1d
I.
Question 1.
Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?
Solution:
Given A {a, b) and B {a, b, C). Yes, A ⊂ B
∴ A ∪ B = {a, b, c) = B.
Question 2.
If A and B are two sets such that A ⊂ B, then what is A ∪ B?
Solution:
Given A and B are two sets such that A ⊂ B then A ∪ B = B.
Question 3.
Which of the following pairs of sets are disjoint.
i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6}.
Solution:
Let A = {1, 2, 3, 4), B = {4, 5, 6).
A ∩ B = {4} ∴ Given pair of sets are not disjoint.
ii) {a, e, i, o, u} and (c, d, e, f}.
Solution:
Let A = {a, e, i, o, u), B = {c, d, e, f)
A ∩ B = {e} ∴ Given pair of sets are not disjoint.
iii) {x: x is an even Integer} and {x: x is an odd Integer}.
Solution:
Let A = {x: x is an even integer), B = {x : x is an odd integer)
A ∩ B = Φ ∴ Given pair of sets are disjoint.
Question 4.
If x = {a, b, c, d} and Y = {f, b, d, g} find
i) X – Y
ii) Y – X
iii) X ∩ Y.
Solution:
Given X = {a,b, c, d}; Y = {f, b, d, g)
i) X – Y = {a, c}.
ii) Y – X = {f, g).
iii) X ∩ Y = (b, d}.
Question 5.
If R is the set of real numbers and 9 is the set of rational numbers, then what is R – Q ?
Solution:
Given, R : set of real numbers.; Q : set of rational numbers.
R – Q = set of irrational numbers.
Question 6.
State whether each of the following statement is true or false. Justify your answer.
i) {2, 3, 4, 5} and {3, 6} are disjoint sets.
ii) {a, e, i, o, u) and {a, b, c, d} are disjoint sets.
iii) (2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
iv) }2, 6, 10} and {3, 7, 11} are disjoint sets.
Solution:
i) False, because 3e { 2, 3, 4, 5} & 3 ∈ {3, 6}.
ii) False, because a € {a, e, i, o, u} & a ∈ (a, b, c, d}.
iii) True, because {2, 6, 10, 14} ∩ {3, 7, 11, 15} = Φ.
iv) True, because {2, 6, 10} ∩ {3, 7, 11} = Φ.
II.
Question 1.
Find the union of each of the following pairs of sets.
i) X = {1, 3, 5}, Y = {1, 2, 3}.
ii) A = [a, e, i, o, u}, B = {a, b, c}.
iii) A = {x : x is a natural number and multiple of 3},
B = {x : x is a natural number less than 6}.
iv) A = {x : x is a natural number and 1 < x < 6},
B = {x : x is a natural number and 6 < x < 10}.
v) A= {1, 2, 3}, B = f
Solution:
i) X = (1, 3, 5}, Y = {1, 2, 3}
X ∪ Y = {1, 3, 5} ∪ {1, 2, 3} = {1, 2, 3, 5}.
ii) A = {a, e, i, o, u}, B = {a, b, c}
∴ A ∪ B = {a, e, i, o, u} ∪ {a, b, c} = {a, b, c, e, i, o, u}.
iii) A = {x : x is a natural number and multiple of 3} = {3, 6, 9, }
B = {x : x is a natural number less than 6} = {1, 2, 3, 4, 5}
A ∪ B = {x : x = 1, 2, 3, 4, 5 and a multiple of 3}.
iv) A = {x : x is a natural number and 1 < x < 6} = {2, 3, 4, 5, 6}
B = {x : x is a natural number and 6 < x < 10} = {7, 8, 9}
A ∪ B = {2,3, 4, 5, 6, 7, 8, 9}.
v) A = {1, 2, 3}, B = Φ; A ∪ B = {1, 2, 3}.
Question 2.
Find the intersection of each pair of sets of question 1 above.
Solution:
i) X = {1, 3, 5}, Y = {1, 2, 3} ⇒ X ∩ Y = {1, 3, 5} ∩ {1, 2, 3} = {1, 3}.
ii) A = {a, e, i, o, u}, B = {a, b, c} A ∩ B = {a}.
iii) A = {3, 6, 9, }, B = {1, 2, 3, 4, 5} ⇒ A ∩ B = {3}.
iv) A = {2, 3, 4, 5, 6}, B = {7, 8, 9} ⇒ A ∩ B = {} = Φ.
v) A = {1, 2, 3}, B = Φ ⇒ A ∩ B = Φ.
Question 3.
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10); find
(i) A ∪ B
(ii) A ∪ C
(iii) B ∪ C
(iv) B ∪ D
(v) A ∪ B ∪ C
(vi) A ∪ B ∪ D
(vii) B ∪ C ∪ D
Solution:
Given A = {.1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}
i) A ∪ B = {1, 2, 3, 4, 5, 6}.
ii) A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}.
iii) B ∪ C = {3,4, 5, 6, 7, 8}.
iv) B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}.
v) A ∪ B ∪ C = {l,2,3, 4, 5, 6} ∪ {5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}.
vi) A ∪ B ∪ D = {1,2, 3, 4, 5, 6} ∪ {7, 8, 9, 10} ={1,2, 3, 4, 5, 6, 7, 8, 9, 10}
vii) B ∪ C ∪ D = {3, 4, 5, 6, 7, 8} ∪ {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8, 9, 10}.
Question 4.
If A = {3. 5, 7. 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17};
find (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)
Solution:
Given A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}
i) A ∩ B = {7, 9, 11}.
ii) B ∩ C = {11, 13}.
iii) A ∩ C ∩ D = (A ∩ C) ∩ D = {11} ∩ {15, 17} = Φ
iv) A ∩ C = {11}.
v) B ∩ D = Φ (or) { }
vi) A ∩ (B ∪ C) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}.
vii) A ∩ D = { } = Φ.
viii) A ∩ (B ∪ D) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15, 17} = {7, 9, 11}.
ix) (A ∩ B) ∩ (B ∪ C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}.
x) (A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 15, 17} ∩ {7,9, 11, 13, 15} = {7, 9, 11, 15}.
III.
Question 1.
If A = {x : x is a natural number}, B = {x : x is an even natural number)
C = {x : x is an odd natural number} and D = [x : x is a prime number},
Find (i) A ∩ B (ii) A ∩ C (iii) A ∩ D (iv) B ∩ C (v) B ∩ D (vi) C ∩ D
Solution:
A = {x : x is a natural number};
B = {x : x is an even natural number}
C = {x ; x is an odd natural number};
D = {x ; x is a prime number}
i) A ∩ B = {x : x is an even natural number} = B.
ii) A ∩ C = {x : x is an odd natural number) = C.
iii) A ∩ D = {x : x is a prime number} = D.
iv) B ∩ C = Φ.
v) B ∩ D = { 2 }.
vi) C ∩ D = {x : x is an odd prime number}.
Question 2.
If 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 (i) A – B (ii) A – C (iii) A – D (iv) B – A (v) C – A (vii) B – C (viii) B – D (ix) C – B (x) D – B (xi) C – D
Solution:
Given 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}.
i) A – B = {3, 6, 9, 15, 18, 21}.
ii) A – C = {3, 9, 15, 18, 21}.
iii) A – D = {3, 6, 9, 12, 18, 21}.
iv) B – A = {4, 8, 16, 20}.
v) C – A = {2, 4, 8, 10, 14, 16}.
vi) D – A = {5, 10, 20}.
vii) B – C = {20}.
viii) B – D = {4, 8, 12, 16}.
ix) C – B = {2, 6, 10, 14}.
x) D – B = {5, 10, 15}.
xi) C – D = {2, 4, 6, 8, 12, 14, 16}.
xii) D – C = {5, 15, 20}.