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4. Cartesian Elimination (15 points) Let A, B, and C be non-empty sets. Prove that (A x B= AXC) → B=C. What happens if A is e

Cartesian elimination. Please prove the statement AND explain the difference if A is empty. my main question is how do I, once I convert the equation to (a in A and b in B)=(a in A and b in B), remove the a in A from both sides? I'm not sure what the proper justification for that is.

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Answer #1

4.

As we are given A X B= A X C.

So, by definition of Cartesian product:

Va € AV E B, and Vc EC,

if (a, b) E AXB , then there also exists the pair (a, b) E AXC .

So, for each b in B, b also exists in C.i.e. all elements in B are also in C.

Similarly we can prove that all c in C also in B.i.e. all elements in C are also in B.

Hence B= C.

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