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Relations — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsRelations5 Marks
Question
If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
$\text{A}\times(\text{B}\cup\text{C})=(\text{A}\times\text{B})\cup(\text{A}\times\text{C})$
✓

Answer

We have,
$\text{A}=\{1,2,3\}\times\{4\}$
$\therefore\ \text{B}\cup\text{C}=\{4\}\cup\{5\}=\{4,5\}$
$\therefore\ \text{A}\times(\text{B}\cup\text{C})=\{1,2,3\}\times\{4,5\}$
$\Rightarrow\text{A}\times(\text{B}\cup\text{C})=\{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)\}\ ...(\text{i}) $
Now,
$\text{A}\times\text{B}=\{1, 2, 3\}\times\{4\}$
$=\{(1, 4), (2, 4), (3, 4)\}$
and, $\text{A}\times\text{C}=\{1,2,3\}\times\{5\}$
$=\{(1, 5) , (2, 5), (3, 5)\}$
$\therefore\ (\text{A}\times\text{B})\cup(\text{A}\times\text{C})=\{(1, 4), (2, 4), (3, 4)\} \cup\{(1, 5), (2, 5), (3, 5)\}$
$\Rightarrow(\text{A}\times\text{B})\cup(\text{A}\times\text{C})=\{(1,4), (1,5), (2, 4), (2, 5), (3, 4), (3, 5)\}\ ...(\text{ii})$
From equation (i) and (ii), we get
$\text{A}\times (\text{B}\cup\text{C})=(\text{A}\times\text{B})\cup(\text{A}\times\text{C})$
Hence verified.

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