0
$\begingroup$

Let $X$, $Y$, and $Z$ be sets of real numbers.

Is it true to say that $(X\cup Y)\cap Z\subset X\cup(Y\cap Z)$?

  • 0
    Strongly related questions: http://math.stackexchange.com/q/435483/11994 and http://math.stackexchange.com/q/544071/11994.2013-11-16

1 Answers 1

2

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 $Y$.

  1. If $x\in X$ then $x\in X\cup(Y\cap Z)$.
  2. If $x\in Y$ then $x\in Y\cap Z$, therefore $x\in X\cup(Y\cap Z)$.

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.

  • 0
    sorry, I was looking at another problem. But thanks, now I'm headed in the right direction2011-10-21