Question

Please answer the above question if u know the answer..

Please
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Answer 1

Let,

$\longrightarrow x\in[(A\cap B^c)^c\cup(B\cap C)]$

Since $x\in (A\cup B)\iff x\in A\lor x\in B,$

$\longrightarrow x\in[(A\cap B^c)^c]\ \lor\ x\in (B\cap C)$

By DeMorgan's Law,

$\longrightarrow x\in(A^c\cup B)\ \lor\ x\in (B\cap C)$

$\longrightarrow (x\in A^c\lor x\in B)\lor(x\in B\land x\in C)$

By distribution law,

$\longrightarrow (x\in A^c\lor x\in B\lor x\in B)\land(x\in A^c\lor x\in B\lor x\in C)$

Since $x\in A\lor x\in A\iff x\in (A\cup A)=A,$

$\longrightarrow (x\in A^c\lor x\in B)\land(x\in A^c\lor x\in B\lor x\in C)$

$\longrightarrow x\in[(A^c\cup B)\cap(A^c\cup B\cup C)]$

Here $(A^c\cup B)\subseteq(A^c\cup B\cup C).$ Then,

$\longrightarrow\underline{\underline{x\in(A^c\cup B)}}$

Hence (B) is the answer.