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Sets — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSSets5 Marks
Question
Let A, B and C be sets. Then show that $\text{A}\cap(\text{B}\cup\text{C})=(\text{A}\cap\text{B})\cup(\text{A}\cap\text{C}).$
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Answer

Let $\text{x}\in\text{A}\cap(\text{B}\cup\text{C})$
$\Rightarrow \text{x}\in\text{A}$ and $\text{x}\in(\text{B}\cup\text{C})$
$\Rightarrow \text{x}\in\text{A}$ and $(\text{x}\in\text{B or x}\in\text{C}) $
$\Rightarrow (\text{x}\in\text{A and x}\in \text{B})$ or $(\text{x}\in\text{A and x}\in\text{C})$
$\Rightarrow \text{x}\in\text{A}\cap\text{B}$ or $\Rightarrow \text{x}\in\text{A}\cap\text{C}$
$\Rightarrow \text{x}\in(\text{A}\cap\text{B})\cup(\text{A}\cap\text{C})$
$\Rightarrow\text{A}\cap(\text{B}\cup\text{C})\subset(\text{A}\cap\text{B})\cup(\text{A}\cap\text{C})$
Now, let $\text{y}\in(\text{A}\cap\text{B})\cup(\text{A}\cap\text{C})$
$\Rightarrow \text{y}\in(\text{A}\cap\text{B})$ or $\text{y}\in(\text{A}\cap\text{C})$
$\Rightarrow (\text{y}\in\text{A and y}\in\text{B})$ or $(\text{y}\in\text{A and y}\in\text{C})$
$\Rightarrow \text{y}\in\text{A}$ and $(\text{y}\in\text{B or y}\in\text{C})$
$\Rightarrow \text{y}\in\text{A}$ and $\text{y}\in\text{B}\cup\text{C}$
$\Rightarrow \text{y}\in\text{A}\cap(\text{B}\cup\text{C})$
$\Rightarrow (\text{A}\cap\text{B})\cup(\text{A}\cap\text{C})\subset\text{A}\cap(\text{B}\cup\text{C})$
Form (i) and (ii), we get.
$\text{A}\cap(\text{B}\cup\text{C})=(\text{A}\cap\text{B})\cup(\text{A}\cap\text{C})$

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