$A \cap (B \cup C) \subseteq (A \cap B) \cup C$ true or false? how to prove?

Question

Drew the Venn diagram for it. Looks like its true to me. But I am confused how I go about proving this. Can someone help please?

Answer 1

It depends on what you are allowed to use, but by the distributive law, $A\cap(B\cup C) = (A\cap B) \cup (A\cap C )$. If $a \in (A\cap B)$ then $a \in (A \cap B) \cup C$ by definition of union, and if $a \in (A \cap C)$ then $a \in C$ by definition of intersection, so $a \in (A \cap B) \cup C$.

Answer 2

If $x$ is an element of $$A\cap (B\cup C)$$ then $x\in A$ and $x\in (B\cup C)$

Therefore either $x\in A$ and $x\in B$ or $x\in A$ and $x\in c$ or both.

We want to show that $$ x\in (A\cap B)\cup C $$

We know that $x\in A$

Now if $x$ is also in B, then $ x\in (A\cap B)$ otherwise $x\in C$ therefore in any case $$ x\in (A\cap B)\cup C $$

Answer 3

Note: $A \cap C \subset C.$

$A \cap (B \cup C)= (A\cap B) \cup (A \cap C)$ (Distributive law)

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