Definition of symmetric relation

Question

I know that the relation is symmetric if $\forall x \forall y \ xRy \implies yRx $.

Consider the set $ A = \{{a, b, c, d}\}$ and $R = \{{(a, a),(a, b),(a, d),(b, a),(b, b),(c, c),(d, a),(d, d)}\}.$

My textbook claims that this relation is symmetric. But what about $(c,d)$ and $(d,c)$ that are not part of the $R$ set? In the definition, it said for all $x$ and $y$, so shouldn't this violate the symmetricity?

Answer 1

If $(c,d)$ is in the relation, then we check whether $(d,c)$ is in the set.

However, in this case, $(c,d)$ is not in $R$, hence we should not expect $(d,c)$ to be in the relation for it to be symmetric.

Answer 2

Definition 1: A relation R over set A is symmetric if for all x, y from A the following is true: (x,y) is in R implies (y,x) is in R.

Definition 2: A relation R over set A is symmetric if for all x, y from A the following is true: if (x,y) is in R, then (y,x) is in R.

These definitions do not require that every (x,y) has to be in R. They only require that for those (x,y) that are in R, (y,x) has to be in R.

Suppose that no single (x,y), where x is different from y, is in R. In this case R is symmetric.

This is so because proposition

  if (x,y) is in R, then  (y,x) is in R

is automatically true when proposition

  (x,y) is in R

is false (i.e. when (x,y) is not in R).