Question

Let U={0, 1, 2, 3, 4, 5, 6, 7}, A={1, 3, 5, 7} and B={0, 2, 3, 5, 7}, find the following sets.
(i) A′ (ii) B′ (iii) A′∪B′ (iv)A′∩B′ ( v)(A∪B)′
(vi) (A∩B)′ (vii) (A′)′ (viii) (B′)′

Answer 1

Solution:-

given by sets:-

U={0, 1, 2, 3, 4, 5, 6, 7}, A={1, 3, 5, 7} and B={0, 2, 3, 5, 7},

(i) A'

》ans :- A' = { 0,2,4,6}

》(i) B'

ans:- B' = { 1,4,6}

(iii) A′∪B′

》ans:- A' = {0, 2,4,6} B' = { 1, 4,6,}

》A′∪B′ = { 0,1,2,4,6}

》(iv) A′∩B'

ans :- A' = {0, 2,4,6} B' = { 1, 4,6,}

》A′∩B' = { 4,6}

(v) (A∪B)′

》ans:- (A∪B) = {0, 1, 2, 3, 4, 5, 6, 7},

》 here U = { x: x is whole numbers}

》 (A∪B)' = {8, 9,10,11,....},

(vi) (A∩B)′

》here A={1, 3, 5, 7} B={0, 2, 3, 5, 7},

ans:-
》 A∩B = {3,5,7} here , U = {x:x is an odd prime numbers },

》(A∩B)′ = { 11,13,17.}

(vii) (A′)′

》ans :-
》 here A' = { 0,2,4,6} , U= { x:x is even numbers}

》then (A′)′ = { 8,10,12,14}

(viii) (B′)′

》ans:-

》here B' = { 1,4,6} U = {0, 1, 2, 3, 4, 5, 6, 7},

》(B′)′ = { 0,2,3,5,6,7}

here , A' and B' are complementary set


Complementary set :- The complement of a set, denoted A', is the set of all elements in the given universal set U that are not in A.

In set- builder notation,  A' = {x ∈ U : x ∉ A}.

☆i hope its help☆