Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

How could I go about proving this?

share|improve this question

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.

share|improve this answer

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 $(A\cap C) \cup (B\cap C)=(A\cup B)\cap C$.

share|improve this answer

Label each of the seven possible sections.

enter image description here

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).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.