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

Question

For any two sets A and B, prove that: $\text{A}\cap\text{B}=\phi\Rightarrow\text{A}\subseteq\text{B}'.$

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Answer

Given $\text{A}\cap\text{B}=\phi,$ i.e., A and B are disjoint set this can represented by venn diagram as follows. To show: $\text{A}\subseteq\text{B}'$ This is clear from the venn diagram it self $\because$ A is lying in the complement of B, but we give a proof of it. So let $\text{x}\in\text{A}$ $\because\text{A}\cap\text{B}=\phi,$ $\therefore\text{x}\not\in\text{B}$ and so $\text{x}\in\text{B}'$ $[\because\text{x}\not\in\text{B}\Rightarrow\text{x}\in\text{B}']$ Thuse $\text{x}\in\text{A}\Rightarrow\text{x}\in\text{B}'.$ This is true for all $\text{x}\in\text{A}$ Hence, $\text{A}\subseteq\text{B}'.$

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