# Thread: Let A,B, and C be sets with A not equal to the empty set. If AXB=AXC, then B=C.

1. ## Let A,B, and C be sets with A not equal to the empty set. If AXB=AXC, then B=C.

Let A,B, and C be sets where A is not equal to the empty set. If AXB = AXC, then B=C.

Assume: AxB=AxC

Prove: B=C

Explain where the assumption that A is not equal to the empty set is needed, in the proof.

I know that I must show that B is a subset of C, and C is a subset of B, therefore sense they are subsets of each other they are equal. But I am not sure at all how to show that they are subsets of each other other than basic assumptions. Any help?

2. ## Re: Let A,B, and C be sets with A not equal to the empty set. If AXB=AXC, then B=C.

Originally Posted by Brjakewa
Let A,B, and C be sets where A is not equal to the empty set. If AXB = AXC, then B=C.
Assume: AxB=AxC Prove: B=C
Start off with the fact that $\left( {\exists a \in A} \right)$ why is that true?
If $b\in B$ then $(a,b)\in A\times B$, WHY?

Thus $b\in C$, WHY?

Now you finish.

If you cannot, then you need a sit-down with a live tutor.