Let $X$, $Y$, and $Z$ be sets of real numbers.
Is it true to say that $(X\cup Y)\cap Z\subset X\cup(Y\cap Z)$?
Let $X$, $Y$, and $Z$ be sets of real numbers.
Is it true to say that $(X\cup Y)\cap Z\subset X\cup(Y\cap Z)$?
Try element chasing:
$x\in (X\cup Y)\cap Z$, then $x\in Z$ and $x\in X\cup Y$, therefore $x\in Z$ and either in $X$ or in $Y$.
Either way we have $x\in (X\cup Y)\cap Z$ then $x\in X\cup(Y\cap Z)$, as wanted. This proof is not just for sets of real numbers but rather for sets in general.