Question

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
(i) $\left(A\cup B\right)\text{'}=A\text{'}\cap B\text{'}$
(ii) $\left(A\cap B\right)\text{'}=A\text{'}\cup B\text{'}$.

Answer 1

Given:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}
We have to verify:

(i) $\left(A\cup B\right)\text{'}=A\text{'}\cap B\text{'}$
LHS
$$\begin{gathered} A\cup B=\{2,3,4,5,6,7,8\} \\ \left(A\cup B\right)\text{'}=\{1,9\} \end{gathered}$$

RHS
$$\begin{gathered} A\text{'}=\{1,3,5,7,9\} \\ B\text{'}=\{1,4,6,8,9\} \\ A\text{'}\cap B\text{'}=\{1,9\} \end{gathered}$$

LHS = RHS
Hence proved.

(ii) $\left(A\cap B\right)\text{'}=A\text{'}\cup B\text{'}$
LHS
$$\begin{gathered} A\cap B=\left\{2\right\} \\ \left(A\cap B\right)\text{'}=\{1,3,4,5,6,7,8,9\} \end{gathered}$$

RHS
$$\begin{gathered} A\text{'}=\{1,3,5,7,9\} \\ B\text{'}=\{1,4,6,8,9\} \\ A\text{'}\cup B\text{'}=\{1,3,4,5,6,7,8,9\} \end{gathered}$$

LHS = RHS
Hence proved.