# Let A and B be relations on the set C = {1,2,3,4,5,6}

## Homework Statement

Let A and B be relations on the set C = {1,2,3,4,5,6}. Prove or disprove the following:

If A and B are symmetric, then A U B is symmetric.

## The Attempt at a Solution

The main problem is that I don't know how A U B is defined.

In general, a relation R is a subset of A x B. For example, {(1,1), (1,2)} is one relation. (ie. I can write 1 R 1, and 1 R 2.)

Symmetric means that for all x, y in C, if x R y then y R x.

I know A U B would be read "A or B", but I don't understand it? Is it like (x A y) or (x B y)?

Then I don't get what A U B would look like if it was symmetric.

Then to prove it you suppose that if x A y then y A x. You also suppose that if x B y then y B x.

From here I don't know what I'm supposed to show...

Dick
Homework Helper

AUB means the union of the sets A and B. It's the union of the set of all of the ordered pairs defining the A relation with all of the ordered pairs defining the B relation. If x AUB y, then either x A y or x B y. Does that help?

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So would it be:

x RUS y if (x R y or x S y) ?

And the condition for symmetry is if x RUS y then y RUS x?

Dick
Homework Helper

So would it be:

x RUS y if (x R y or x S y) ?

And the condition for symmetry is if x RUS y then y RUS x?

Exactly.

Perfect, thanks!