For any two sets A and B, prove that: A' - B' = B - A. - Vidyadip

Question

For any two sets A and B, prove that: A' - B' = B - A.

✓

Answer

To show $\text{A}' - \text{B}' = \text{B} - \text{A}$
We show that $\text{A}' – \text{B}' = \subseteq\text{B} - \text{A} $ and vice versa
Let, $\text{x}\in\text{A}'-\text{B}'$
$\Rightarrow\text{x}\in\text{A}'\text{and x}\not\in\text{B}'$
$\Rightarrow\text{x}\not\in\text{A }\text{and x}\in\text{B}$ $[\because\text{A}\cap\text{A}'=\oint\text{ and B}\cap\text{B}'=\oint]$
$\Rightarrow\text{x}\in\text{B}\text{ and x}\not\in\text{A}$
$\text{x}\in\text{B}- \text{A}$
This is true for all $\text{x}\in\text{A}'-\text{B}'$
Hence $\text{A}'-\text{B}'\subseteq\text{B}-\text{A}$
Conversely,
Let, $\text{x}\in\text{B} - \text{A}$
$\Rightarrow\text{x}\in\text{B and x}\not\in\text{A}$
$\Rightarrow\text{x}\not\in\text{B}'\text{ and x}\in\text{A}'$
$\Rightarrow\text{x}\in\text{A}'\text{ and x}\not\in\text{B}'$ $[\because\text{B}\cap\text{B}'=\oint\text{ and A}\cap\text{A}'=\oint]$
$\Rightarrow\text{x}\in\text{A}'-\text{B}'$
This is true for all $\text{x}\in\text{B} - \text{A}$
Hence $\text{B}- \text{A}\subseteq\text{A}'=\text{B}'$
$\therefore\text{ A}' - \text{B}' = \text{B} - \text{A}$ Proved.

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