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Question Number 26889 by shiva123 last updated on 31/Dec/17

Answered by prakash jain last updated on 31/Dec/17

$$3.A\cap (B-C)=(A\cap B)-(A\cap C)$$$$x\in A\cap (B-C)$$$$\Rightarrow x\in A\wedge x\in (B-C)$$$$\Rightarrow x\in A\wedge x\in B\wedge x\notin C$$$$\Rightarrow x\in (A\cap B)\wedge x\notin (A\cap C)$$$$\Rightarrow x\in (A\cap B)-(A\cap C)$$$$\Rightarrow (A\cap B)-(A\cap C)\supseteq A\cap (B-C)\left(I\right)$$$$x\in (A\cap B)-(A\cap C)$$$$\Rightarrow x\in (A\cap B)\wedge x\notin (A\cap C)$$$$\Rightarrow x\in A\wedge x\in B\wedge x\notin (A\cap C)$$$$\Rightarrow x\in A\wedge x\in B\wedge x\notin C$$$$\Rightarrow x\in A\wedge x\in (B-C)$$$$\Rightarrow x\in A\cap (B-C)$$$$\Rightarrow (A\cap B)-(A\cap C)\subseteq A\cap (B-C)\left(II\right)$$$$\mathrm{From}\left(I\right)\mathrm{and}\left(II\right)$$$$(A\cap B)-(A\cap C)=A\cap (B-C)◼$$

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