# Cross Products and Set Unions Proof

• August 27th 2012, 01:50 PM
renolovexoxo
Cross Products and Set Unions Proof
I'm hoping this is the right category, since it's set theory. If it's not I'll move it if someone knows a better place for these questions

Prove that A X (B1U B2) = (A X B1) U (A X B2)

I am actually stuck on the B1 B2 in this problem. It's some notation aspect that is keeping me from understanding what the problem is asking for. Are B1 and B2 separate sets?
• August 27th 2012, 02:12 PM
emakarov
Re: Cross Products and Set Unions Proof
Quote:

Originally Posted by renolovexoxo
Are B1 and B2 separate sets?

Yes, B1 and B2 are arbitrary sets. The problem is asking to prove that A x (B U C) = (A x B) U (A x C) for arbitrary sets A, B, C.
• August 27th 2012, 02:24 PM
Plato
Re: Cross Products and Set Unions Proof
Quote:

Originally Posted by renolovexoxo
I'm hoping this is the right category, since it's set theory. If it's not I'll move it if someone knows a better place for these questions

Prove that A X (B1U B2) = (A X B1) U (A X B2)

\begin{align*}(x,y)\in A\times (B\cup C)&\text{if and only if }x\in A\text{ and }y\in (B\cup C) \\&\text{if and only if }x\in A\text{ and }(y\in B\text{ or }y\in C) \\&\text{if and only if }[x\in A\text{ and }(y\in B)]\text{ or }[ x\in A\text{ and }(y\in C))] \\&\text{if and only if }(x,y)\in(A\times B)\text{ or }(x,y)\in(A\times C)\end{align*}