Fill in the blanks:
(i) If A, B are two sets, then A ∪ B = .... .
(ii) If A, B are two sets, then A ∩ B = .... .
(iii) If A, B are two sets, then A - B = .... .
(iv) A and B are disjoint sets if and only if A ∩ B = .... .
(v) A and B are overlapping sets if and only if A ∩ B .... .
(vi) The set {x : x ∈ W, x < 3} in the roster form = .... .
(vii) If A is any set, then A ∪ φ = .... and A ∪ ξ = .... .
(viii) If ξ = {all digits in our number system} and A = {1, 2, 3, 4, 5}, then A′ = .... .
(ix) If A is any set and A′ is its complement, then A ∪ A′ = .... and A ∩ A′ = .... .
Answer
(i) If A, B are two sets, then A ∪ B = {x | x ∈ A or x ∈ B}.
(ii) If A, B are two sets, then A ∩ B = {x | x ∈ A and x ∈ B}.
(iii) If A, B are two sets, then A - B = {x | x ∈ A and x ∉ B}.
(iv) A and B are disjoint sets if and only if A ∩ B = φ.
(v) A and B are overlapping sets if and only if A ∩ B ≠ φ.
(vi) The set {x : x ∈ W, x < 3} in the roster form = {0, 1, 2}.
(vii) If A is any set, then A ∪ φ = A and A ∪ ξ = ξ.
(viii) If ξ = {all digits in our number system} and A = {1, 2, 3, 4, 5}, then A′ = {0, 6, 7, 8, 9}.
(ix) If A is any set and A′ is its complement, then A ∪ A′ = ξ and A ∩ A′ = φ.
State whether the following statements are true (T) or false (F):
(i) If ξ is the universal set and A is any set, then A′ = {x; x ∈ ξ and x ∉ A}
(ii) If A = {0, 1, 2, 3, 4, 5} and B = {0, 3, 5, 7}, then A ∩ B = B
(iii) If A = {0, 1, 2, 3, 4, 5} and B = {0, 3, 5, 7}, then A ∪ B = A
(iv) If ξ = {all non-zero digits in our number system}, A = {multiples of 2} and B = {multiples of 3}, then A ∩ B = {6}.
Answer
(i) True
The complement of a set A is the set consisting of all those elements of ξ which do not belong to A, i.e., A′ = {x; x ∈ ξ and x ∉ A}.
(ii) False
A ∩ B = {0, 1, 2, 3, 4, 5} ∩ {0, 3, 5, 7} = {0, 3, 5}
As 7 ∈ B but 7 ∉ A ∩ B,
∴ A ∩ B ≠ B
(iii) False
A ∪ B = {0, 1, 2, 3, 4, 5} ∪ {0, 3, 5, 7} = {0, 1, 2, 3, 4, 5, 7}
As 7 ∈ A ∪ B but 7 ∉ A,
∴ A ∪ B ≠ A
(iv) True
ξ = {all non-zero digits in our number system} = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {multiples of 2} = {2, 4, 6, 8}
B = {multiples of 3} = {3, 6, 9}
∴ A ∩ B = {2, 4, 6, 8} ∩ {3, 6, 9} = {6}