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

Question

For any two sets of A and B, prove that:$\text{B}'\subset\text{A}'\Rightarrow\text{A}\subset\text{B.}$

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Answer

We have $\text{B}'\subset\text{A}'$ To show: $\text{A}\subset\text{B}$ Let, $\text{x}\in\text{A}$ $\Rightarrow\text{x}\not\in\text{A}'$ $[\because\text{A}\cap\text{A}'=\phi]$ $\Rightarrow\text{x}\not\in\text{B}'$ $[\because\text{B}'\subset\text{A}']$ $\Rightarrow\text{x}\in\text{B}$ $[\because\text{B}\cap\text{B}'=\phi]$ Thus, $\text{x}\in\text{A}\Rightarrow\text{x}\in\text{B}$ This is true foe all $\text{x}\in\text{A}$ $\therefore\text{ A}\subset\text{B}.$

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