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

Question

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

✓

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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