For any two sets A and B, prove the following: A - B=A (A ) - Vidyadip

Question

For any two sets A and B, prove the following: $\text{A} - \text{B}=\text{A }\Delta\text{ (A}\cap\text{B})$

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Answer

$\text{RHS}=\text{A }\Delta\text{ (A} \cap8)$ $=\text{(A} - \text{(A}\cap\text{B}))\cup\text{(A}\cap\text{B} - \text{A})$ $[\because\text{E }\Delta\text{ F} =\text{(E} - \text{F)}\cup\text{(F} - \text{E)]}$ $=\text{(A}\cap\text{(A}\cap\text{B})')\cup\text{(A}\cap\text{B}\cap\text{A}')$ $[\because \text{E - F = E }\cap \text{F}']$ $=\text{(A}\cap \text{(A}'\cup\text{B}'))\cup\text{(A}\cap\text{A}'\cap\text{B})$ [By de-morgan's law & associative law] $=\text{(A}\cap\text{A}')\cup(\text{A}\cap\text{B}')\cup(\oint\cap\text{B})$ $[\because\cap\text{ distributes over}\cup\text{and A }\cap \text{A}' = \oint]$ $= \oint \cup \text{(A}\cap\text{B}')\cup \oint$ $[\because \oint \cap \text{ B} = \oint]$ $= \text{A}\cap\text{B}'$ $[\because\oint\cup\text{ x = x for any set x}]$ $= \text{A} - \text{B }$ $[\because \text{A} \cap\text{B}' = \text{A - B}]$ $=\text{LHS}$ $\therefore$ LHS = RHS Proved.

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