If ξ = {x : x ∈ N, x ≤ 10 } , A = {x : x ≥ 5} and B = {x : 3 < x < 6}, then find :

(i) (A ∪ B)'

(ii) A' ∩ B'

Are (A ∪ B)' and A' ∩ B' equal ?

ξ = {x : x ∈ N, x ≤ 10 }

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {x : x ≥ 5}

A = {5, 6, 7, 8, 9, 10}

B = {x : 3 < x < 6}

B = {4, 5}

(i) (A ∪ B)'

A ∪ B - contains all the elements in set A and B.

A ∪ B = {5, 6, 7, 8, 9, 10} ∪ {4, 5}

A ∪ B = {4, 5, 6, 7, 8, 9, 10}

(A ∪ B)'- contains all the elements in universal set which are not in A ∪ B.

(A ∪ B)' = ξ - A ∪ B

(A ∪ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {4, 5, 6, 7, 8, 9, 10}

(A ∪ B)' = {1, 2, 3} ……………(1)

(A ∪ B)' = {1, 2, 3}

(ii) A' ∩ B'

A' - contains all the elements in universal set which are not in A.

A' = ξ - A

A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {5, 6, 7, 8, 9, 10}

A' = {1, 2, 3, 4}

B' - contains all the elements in universal set which are not in B.

B' = ξ - B

B' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {4, 5}

B' = {1, 2, 3, 6, 7, 8, 9, 10}

A' ∩ B' - contains all the common elements in set A and B.

A' ∩ B' = {1, 2, 3, 4} ∩ {1, 2, 3, 6, 7, 8, 9, 10}

A' ∩ B' = {1, 2, 3} ……………(2)

A' ∩ B' = {1, 2, 3}

From (1) and (2) , we can see that

Yes, (A ∪ B)' and A' ∩ B' are equal.