If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
$\text{A}\times(\text{B}\cap\text{C})=(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$
✓
Answer
We have,
$\text{A}=\{1,2,3\},\text{ B}=\{4\}$ and $\text{C}=\{5\}$
$\therefore\ \text{B}\cap\text{C}=\{4\}\cap\{5\}=\phi$
$\therefore\ \text{A}\times(\text{B}\cap\text{C})=\{1,2,3\}\times\phi$
$\Rightarrow\text{A}\times(\text{B}\cap\text{C})=\phi\ ...(\text{i})$
Now,
$\text{A}\times\text{B}=\{1, 2, 3\}\times\{4\}$
$=\{(1, 4), (2, 4), (3, 4)\}$
and, $\text{A}\times\text{C}=\{1,2,3\}\times\{5\}$
$=\{(1, 5) , (2, 5), (3, 5)\}$
$\therefore\ (\text{A}\times\text{B})\cap(\text{A}\times\text{C})=\{(1, 4), (2, 4), (3, 4)\} \cup\{(1, 5), (2, 5), (3, 5)\}$
$\Rightarrow(\text{A}\times\text{B})\cup(\text{A}\times\text{C})=\phi\ ...(\text{ii})$
From equation (i) and (ii), we get
$\text{A}\times (\text{B}\cap\text{C})=(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$
Hence verified.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.