Question

If U = {x : x $ϵ$ Z, -3 < x $\le$ 9}, A = {x : x = 2P + 1, P $ϵ$ Z , -2 $\le$ P $\le$ 3}, B = {x : x = q + l, q $ϵ$ Z, 0 $\le$ q $\le$ 3}, verify De Morgan’s laws for complementation.

Answer 1

Given, U = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {-3, -1, 1, 3, 5, 7} and B = {1, 2, 3, 4}

Law (i) (A $\cup$ B)' = A' $\cap$ B'

Now, A $\cup$ B = {-3, -1, 1, 2, 3, 4, 5, 7}

(A $\cup$ B)' = {-2, 0, 6, 8, 9}...(1)

Then, A' = {-2, 0, 2, 4, 6, 8, 9) and

B' = {-3, -2, -1, 0, 5, 6, 7, 8, 9}

A' $\cap$ B' = {-2, 0, 6, 8, 9}...(2)

From (1) and (2) it is verified that

(A $\cup$ B)' =A' $\cap$ B'

Law (ii) (A $\cap$ B)' = A' $\cup$ B'

Now, A $\cap$ B = {1, 3}

(A $\cap$ B)' = {-3, -2, -1, 0, 2, 4, 5, 6, 7, 8, 9}...(3)

Then, A' $\cup$ B' = {-3, -2, -1, 0, 2, 4, 5, 6, 7, 8, 9}...(4)

From (3) and (4) it is verified that

(A $\cap$ B)' = A' $\cup$ B'