Let us suppose a function $f \colon \mathbb{N} \times \mathbb{N} \to \mathbb{R}$, such that $$\neg\left(\forall a,b \,|\, a \in N \land b \in N \implies f(a,b)=f(b,a)\right)$$ That is $$\left(\exists a,b \,|\, a \in N \land b \in N \land f(a,b)\neq f(b,a)\right)$$ I'm not sure if I can say that f is non-symmetric. Can I? I know that a I can use symmetric, asymmetric, anti-symmetric for characterizing relations. But I think that when I'm working with functions, the reasoning is not the same.
Thank you in advance for your attention.
Best regards.