# Thread: [SOLVED] set proof help

1. ## [SOLVED] set proof help

Can anyone prove or disprove
$X \cap Z \subseteq Y \Leftrightarrow (X - Y) \cup (Y-Z) \subseteq Z^c$

I think you have to split it up into two parts, but whenever I do it one way I can't get the same exact argument to go back the other way.

2. Originally Posted by horan
Can anyone prove or disprove
$X \cap Z \subseteq Y \Leftrightarrow (X - Y) \cup (Y-Z) \subseteq Z^c$

I think you have to split it up into two parts, but whenever I do it one way I can't get the same exact argument to go back the other way.
Hi

Suppose that $X \cap Z \subseteq Y$
Let's prove that $(X - Y) \cup (Y-Z) \subseteq Z^c$

Let $x \in (X - Y) \cup (Y-Z)$
We have to prove that $x \in Z^c$

$x \in (X - Y) \cup (Y-Z)$ means that or $x \in (Y-Z)$ or $x \in (X - Y)$
If $x \in (Y-Z)$ then $x \notin Z$ and therefore $x \in Z^c$ => OK
If $x \in (X - Y)$ then $x \in X$ and $x \notin Y$. If $x \in Z$ then $x \in X \cap Z$. But $X \cap Z \subseteq Y$ therefore $x \in Y$ which is not possible since $x \notin Y$. Threfeore $x \notin Z$.
In both cases $x \in Z^c$.
Therefore $(X - Y) \cup (Y-Z) \subseteq Z^c$.

3. ## Set Theory Proof

Hello horan
Originally Posted by horan
Can anyone prove or disprove
$X \cap Z \subseteq Y \Leftrightarrow (X - Y) \cup (Y-Z) \subseteq Z^c$

I think you have to split it up into two parts, but whenever I do it one way I can't get the same exact argument to go back the other way.
running-gag has proved that $X \cap Z \subseteq Y \Rightarrow (X - Y) \cup (Y-Z) \subseteq Z^c$. Here's the proof of $X \cap Z \subseteq Y \Leftarrow (X - Y) \cup (Y-Z) \subseteq Z^c$.

Suppose that $(X - Y) \cup (Y-Z) \subseteq Z^c$and that $x \in X\cap Z$. Therefore we must show that $x \in Y$.

$x \in X \cap Z \Rightarrow x \in Z$

$\Rightarrow x \notin Z^c$

$\Rightarrow x \notin (X - Y) \cup (Y-Z)$

$\Rightarrow x \notin (X - Y)$

But $x \in X$

Therefore $x \in Y$

4. hello running-gag
..If then ..
how can you say like that..

5. Hello doresa
Originally Posted by doresa
hello running-gag

..If then ..

how can you say like that..
You're just quoting part of the whole proof here. Obviously you can't say that, in general, $x \in Z \Rightarrow x \in X \cap Z$.

But the bit you're quoting is from a line that starts: If $x \in (X-Y)$... In other words, it pre-supposes that $x \in X$. Therefore if in addition $x \in Z$, then $x \in X \cap Z$.

Is that OK now?