Why $R=\{(a,b)\mid a=b \mbox{ or } a=-b\}$ is not anti symmetric?

Question

Why $R=\{(a,b)\mid a=b \mbox{ or } a=-b\}$ is not anti symmetric?

I read because this is symmetric so it is not anti symmetric, but $R=\{(a,b) \mid a=b \}$ is both symmetric and anti symmetric.

Answer 1

It’s not antisymmetric because $\langle 1,-1\rangle\in R$ and $\langle -1,1\rangle\in R$, but $1\ne -1$. That’s a clear violation of the definition of antisymmetry. Remember, antisymmetry means that if $\langle a,b\rangle\in R$ and $\langle b,a\rangle\in R$, then $a=b$.

Note also that it is entirely possible for a relation to be both symmetric and antisymmetric. The relation of equality on any set is both symmetric and antisymmetic.

Answer 2

Suppose $X=\{-1,1\}$. We see that $R(-1,1)$ and $R(1,-1)$ which violates $R$ being an antisymmetric relation because $1\neq -1$.