Language concatenation property

Question

How i can prove that

A$\cdot$(B$\cap$C) $\subseteq$ A$\cdot$B $\cap$ A$\cdot$C

Where $\cdot$ is the language concatenation operator. I know that the inverse is false. Thanks

Answer 1

First, recall the formal definition of the intersection : we say that $w\in A\cap B\Leftrightarrow w\in A$ and $w\in B$.

Now, suppose say we have a word $w\in A\cdot (B\cap C)$. We know we can write $w=uv$, where $u\in A$ and $v\in B\cap C$. But being in $B\cap C$ means that $v\in B$ and $v\in C$. Then, we can say that $uv\in A\cdot B$, and similarly, $uv\in A\cdot C$. As $w=uv$ is in both $A\cdot B$ and $A\cdot C$, it is in the intersection : $w\in (A\cdot B)\cap (A\cdot C)$.

Now, since this works for any $w\in A\cdot (B\cap C)$, we can say that $A\cdot (B\cap C)\subseteq(A\cdot B)\cap (A\cdot C)$.

Answer 2

Hint. First show that if $X \subseteq Y$, then $ZX \subseteq ZY$.