(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For sets A and B, define a relation [tex]\mathcal{R}[/tex] on A∪B by:

[tex]\forall A, B \in A \cup B, x\mathcal{R}y[/tex] if and only if [tex](x,y) \in A \times B[/tex]

For all sets A and B, if R is symmetric, then A = B

2. Relevant equations

3. The attempt at a solution

I tried doing this, and I heard it's supposed to be false. BUT I can't see why it can be. The cartesian product should emply that x and y are only related when x is in A and y is in B. So, if it's symmetric, then it's already assumed that all elements in A are related to an element in B, and similarly, all elements in B are related to that same element in A. Which implies that they are equal. No?

Is my logic messed up? I don't see how this could be false. If either of A or B were the empty set, there'd be nothing to prove, since the cartesian product of anything with an empty set is empty, then the if part can never be assumed, and it should be vacuously true.

Can anyone point me in the right direction if I'm doing this wrong? Thanks!

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# Homework Help: Symmetric relation on ordered pairs

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