Question

Show that the relation R on a set A is symmetric if and only if R = R^−1, where R^−1 is the inverse relation.

I was wondering if we could show this using a matrix. We know that being symmetric implies a matrix has to have a full diagonal entries of 1. Therefore, the inverse of an identity matrix ( lets say In) would be identical, thus proving the relationship. I was wondering If that would be right? any help?

Answer 1

That is just difficult.

Answer 2

I don't need to know if its difficult or not. I don't need to have comments like that on my question. It is a legit discrete math question, very basic actually, and using normal definitions should be fine. I was asking for confirmation on my train of thought, but thankyou for your input.

Answer 3

Choose any (x,y)∈R∩R−1. we can show that R∩R−1 is symmetric, if it suffices to show that (y,x)∈R∩R−1. Since (x,y)∈R∩R−1, we know that (x,y)∈R and (x,y)∈R−1. The earlier implies that (y,x)∈R−1 and the latter implies that (y,x)∈R. Hence, we conclude that (y,x)∈R∩R−1, as desired.