Symmetric relations

Question

Let $A=\{ 1, 2, 3, 4 \}$ is $ B = \{ (1, 2), (2, 1), (1, 3), (3, 1) \} $

Is $B$ a symmetric relation on $A$?

I said no because not all $x, y \in A$ are in $B$

Is this correct?

Answer 1

No, $B$ is a symmetric relation on $A$, because since if $(a,b) \in B$ then $(b,a) \in B$.

Answer 2

The definition of symmetric relation does not really require us to have any $A$ to begin with.

We say that $R$ is a symmetric relation if the following holds: Whenever an ordered pair $(a,b)\in R$ then the reversed pair, $(b,a)\in R$ as well.

Symmetry, as well as transitivity, are internal properties of a relation (in contrast of "reflexivity" which is indeed external).

Finally, note that under your "definition" the only symmetric relation on $A$ would be $A\times A$. That's not a very interesting property now, is it?