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How would one prove the following distributive laws for the following:

1) A∪(B∩C)=(A∪B)∩(A∪C)

2) A∩(B∪C)=(A∩B)∪(A∩C)

We are given an example proof that the functions are associative:

A∪(B∪C)=A∪{x|x∈B∨x∈C}

        ={x|x∈A∨(x∈B∨x∈C)}

        ={x|(x∈A∨x∈B)∨x∈C}

        =(A∪B)∪C

1 Answers 1

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Just follow the example you are given. I'm doing the first one, the other is similar. \begin{equation*} \begin{aligned} A \cup (B \cap C)&= A \cup \{x:x \in B \land x \in C\} \\ &= \{ x : (x \in A) \lor (x \in B \land x \in C) \} \\ &= \{ x : (x \in A \lor x \in B) \land (x \in A \lor x \in C) \} \\ &= (A \cup B)\cap(A \cup C). \\ \end{aligned} \end{equation*}