Cartesian product of universal set.

**truevein** · November 6th 2010, 12:10 PM

Hi, I don't know how to prove this.

$UXA^C = (UXA)^C$

What is the result of UxA? I know AxØ=Ø but have no idea about universal set. Thanks for any help.

**Drexel28** · November 6th 2010, 12:21 PM

Originally Posted by truevein

Hi, I don't know how to prove this.

$UXA^C = (UXA^C)$

What is the result of UxA? I know AxØ=Ø but have no idea about universal set. Thanks for any help.

I believe you're trying to prove that $U\times\left(U-A\right)=U-\left(U\times A\right)$ . Is that right?

**truevein** · November 6th 2010, 12:39 PM

Really sorry about the displacement of the parenthesis, I corrected my question.

**Plato** · November 6th 2010, 12:56 PM

We know that $\left( {\forall x} \right)\left[ {x \in U} \right]$ , the universe.

So if $(x,y)\in U\times A^c$ then because $x\in U$ we know $y\notin A$ so $(x,y)\in (U\times A)^c$ .

If $(u,w)\in (U\times A)^c$ then $u \notin U \vee w \notin A$ .
Again we know that $u\in U$ so $w\notin A$ .
So $(u,w)\in U\times A^c$ .

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