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 (B_{1}U B_{2}) = (A X B_{1}) U (A X B_{2})

I am actually stuck on the B_{1} B_{2} in this problem. It's some notation aspect that is keeping me from understanding what the problem is asking for. Are B_{1} and B_{2} separate sets?

Re: Cross Products and Set Unions Proof

Quote:

Originally Posted by

**renolovexoxo** Are B_{1} and B_{2} separate sets?

Yes, B_{1} and B_{2} 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.

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 (B_{1}U B_{2}) = (A X B_{1}) U (A X B_{2})

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