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 P: 492 Let $P(n)$ be $$\left(A_1\,\cap\,A_2\,\cap\,\dots\,\cap\,A_n\right)\,\cup\,B\,=\,\left( A_1\,\cup\,B\left)\,\cap\,\left(A_2\,\cup\,B\right)\,\cap\,\dots\,\cap\ ,\left(A_n\,\cup\,B\right)$$ Then $\forall\,n\,\left(P(n)\right)$, right? First I do the basis step for $P(1)$: $$A_1\,\cup\,B\,=\,\left(A_1\,\cup\,B\right)$$ Now I need to show that $P(k)\,\longrightarrow\,P(k\,+\,1)$? For $P(k)$ (Also, assume it is true for induction): $$\left(A_1\,\cap\,A_2\,\cap\,\dots\,\cap\,A_k\right )\,\cup\,B\,=\,\left(A_1\,\cup\,B\left)\,\cap\,\left(A_2\,\cup\,B\right )\,\cap\,\dots\,\cap\,\left(A _k\,\cup\,B\right)$$ EDIT: Removed errors