If A = {2, 3}, B = {4, 5}, C = {5, 6}, find $\text{A}\times(\text{B}\cap\text{C}),\text{ A}\times(\text{B}\cap\text{C}),(\text{A}\times\text{B})\cup(\text{A}\times\text{C}).$
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Answer
We have,
$\text{A}=\{2,3\},\text{ B}=\{4,5\}$ and $\text{C}=\{5,6\}$
$\therefore\ \text{B}\cup\text{C}=\{4,5\}\cup\{5,6\}=\{4,5,6\}$
$\therefore\ \text{A}\times\{\text{B}\cup\text{C}\}=\{2,3\}\times\{4,5,6\}$
$= \{(2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)\}$
Now,
$\text{B}\cap\text{C}=\{4,5\}\cap\{5,6\}=\{5\}$
$\therefore\ \text{A}\times(\text{B}\cap\text{C})=\{2,3\}\times\{5\}$
$\text{A}\times(\text{B}\cap\text{C})=\{(2,5),(3,5)\}$
Now,
$\text{A}\times\text{B}=\{2,3\}\times\{4,5\}$
$= \{(2,4), (2,5), (3, 4), (3, 5)\}$
and $\text{A}\times\text{C}=\{2,3\}\times\{5,6\}$
$=\{(2, 5), (2, 6), (3, 5), (3, 6)\}$
$\therefore\ (\text{A}\times\text{B})\cup(\text{A}\times\text{C})= \{(2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)\}$
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