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

$\left(i\right)$ Given: $U=\{1,2,3,4,5,6,7,8,9\},A=\{2,4,6,8\}$ and $B=\{2,3,5,7\}$

Step 1: Finding $(A\cup B{)}^{\prime }$
$A\cup B=\{2,3,4,5,6,7,8\}$
So, $(A\cup B{)}^{\prime }=U-(A\cup B)$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,3,4,5,6,7,8\}$
$\Rightarrow (A\cup B{)}^{\prime }=\{1,9\}$

Step 2: Finding $A^{\prime }\cap B^{\prime }$
We know, $A=\{2,4,6,8\}$ and $B=\{2,3,5,7\}$
$\therefore A^{\prime }=U-A$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\}$
$=\{1,3,5,7,9\}$
And $B^{\prime }=U-B$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,3,5,7\}$
$=\{1,4,6,8,9\}$
$\therefore A^{\prime }\cap B^{\prime }=\{1,3,5,7,9\}\cap \{1,4,6,8,9\}$
$\Rightarrow A^{\prime }\cap B^{\prime }=\{1,9\}$

Final answer:
Hence verified $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$

$\left(ii\right)$ Given: $U=\{1,2,3,4,5,6,7,8,9\},A=\{2,4,6,8\}$ and $B=\{2,3,5,7\}$

Step 1: Finding $(A\cap B{)}^{\prime }$
$A\cap B=\{2,4,6,8\}\cap \{2,3,5,7\}=\left\{2\right\}$
Now, $(A\cap B{)}^{\prime }=\cup -(A\cap B)=\{1,2,3,4,5,6,7,8,9\}-\left\{2\right\}$
$\Rightarrow (A\cap B{)}^{\prime }=\{1,3,4,5,6,7,8,9\}$

Step 2: Finding $A^{\prime }\cup B^{\prime }$
$A^{\prime }=U-A$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\}=\{1,3,5,7,9\}$
And $B^{\prime }=U-B$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,3,5,7\}=\{1,4,6,8,9\}$
$\therefore A^{\prime }\cup B^{\prime }=\{1,3,5,7,9\}\cup \{1,4,6,8,9\}$
$A^{\prime }\cup B^{\prime }=\{1,3,4,5,6,7,8,9\}$

Final answer:
Hence verified, $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$