So I am learning about proving intersection and union statements of sets, but the problem is I am never confident about my proofs, I never know when I am right. So if you could check my attempt, and maybe offer some help that would be great:
Prove: $A\cup \!\, (B\cap \!\ C)=(A\cup \!\, B)\cap \!\ (A\cup \!\, C)$
Here's my attempt:
$A\cup \!\, (B\cap \!\ C)=A\cup \!\,${$x:x\in \!\, B \ \text{and}\ x\in \!\, C $}
$=${$x:x\in \!\, A \ \text{or}\ x\in \!\, C\ \text{and} \ x\in \!\, B $}
(I think I need a step in between here, right?)
$=(A\cup \!\, B)\cap \!\ (A\cup \!\, C)$