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Prove: $(A\cup B) \cap C \subseteq A\cup (B\cap C)$

How could I go about proving this?

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3 Answers 3

up vote 6 down vote accepted

Suppose $x \in (A\cup B) \cap C$. Then $x\in C$ and $x\in A\cup B$.

Either $x\in A$ or $x\in B$. If it is in $A$ then it is in $A\cup (B\cap C)$. If it is in $B$ then it is in $B\cap C$, and we are done.

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One possible way is to rewrite it as $$ [(x\in A \lor x\in B)\land x\in C] \Rightarrow [x \in A \lor (x\in B \land x\in C)]$$ and to verify that $[(p\lor q)\land r] \Rightarrow [p\lor (q\land r)]$ is a tautology.

Another way is using Venn diagrams. (See a little don's answer or wikipedia.)

Another possibility is use some facts you've already learned, like: $A\cap C\subseteq A$ implies $(A\cap C) \cup (B\cap C)\subseteq A\cup (B\cap C)$ and $(A\cap C) \cup (B\cap C)=(A\cup B)\cap C$.

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Label each of the seven possible sections.

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Determine which of those are in (A ∪ B) ∩ C, then which are in A ∪ (B ∩ C).

Every section on the list for (A ∪ B) ∩ C should be on the list for A ∪ (B ∩ C).

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