Can I make the following assumption about symmetric relations?

Question

If $R$ is a symmetric relation then:

$$(x,y) \not \in R \rightarrow (y,x) \not \in R$$

Answer 1

Yes; the statements are equivalent.

Symmetry is the property that: $\forall x, y: \big((y,x)\in R\to (x,y)\in R\big)$

The contraposition of this is: $\forall x, y:\big( (x,y)\notin R\to (y,x)\notin R\big)$