Is the following proposition true or false? Justify your conclusion.
Let A, B, and C be sets with A not equal to the empty set. If the Cartesian Product AXB = the Cartesian Product AXC, then B=C.
Explain where the assumption that A is not equal to the empty set is needed.
This proposition seems true to me, but I have no idea how to prove it. I tried showing that AXB = AXC by showing they are subsets of each other, but I'm not sure if that's the right approach. Additionally, my proof of showing they are subsets of one another didn't get very far.
Can anyone show me how to do this step by step?