[SOLVED] set proof help

**horan** · March 5th 2009, 10:30 AM

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** · March 5th 2009, 10:52 AM

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 $(X - Y) \cup (Y-Z) \subseteq Z^c$ .

**Grandad** · March 6th 2009, 02:41 AM

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 $x \in Y$

Grandad

**doresa** · July 17th 2009, 07:43 AM

hello running-gag

..If

then

..

how can you say like that..

**Grandad** · July 17th 2009, 10:01 AM

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?

Grandad

**doresa** · July 18th 2009, 08:48 AM

ok.thank you

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