Question

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

Answer 1

(i) Compute that $(A\cup B)\text{'}=A\text{'}\cap B\text{'}$

We are given, $U=\{1,2,3,4,5,6,7,8,9\},A=\{2,4,6,8\}$ and $B=\{2,3,5,7\}$

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

Also

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

On solving,

$A\text{'}\cap B\text{'}=\{1,9\}$

Therefore,$(A\cup B)\prime =A\prime \cap B\prime$, proved.

(ii) Compute that $(A\cap B)\prime =A\prime \cup B\prime$

We have,

$\left(A\cap B\right)=\left\{2\right\}$

That is

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

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

So we get,$(A\cap B)\prime =A\prime \cup B\prime .$

Hence, proved.

(I)$(A\cup B)\prime =A\prime \cap B\prime$

(ii)$(A\cap B)\prime =A\prime \cup B\prime .$