How can a relation be both not symmetric and not antisymetric?

Question

I am aware that relations can be both symmetric and antisymmetric, or either one of the two. However, I am still a little bit confused as to why they can not be both (i.e. not symmetric and not antisymmetric)?

Answer 1

On the set $\{1,2,3\}$, take the relation $$ \{(1,2),(2,1),(1,3)\} $$ It has $(1,3)$ but not $(3,1)$ so it isn't symmetric. Also, it has both $(1,2)$ and $(2,1)$, but $1\neq 2$ so it isn't antisymmetric.

Answer 2

it's just a bad terminology. If $Sym$ is a symmetric relation on the set $X$ then the set difference $X \times X \setminus Sym$ not necessarily to be an antisymmetric relation

Answer 3

Sure, they can be both.

The most common non-symmetric relations in mathematics come from partial orders. Most of the time, that's going to give us an antisymmetric relation, but it doesn't have to. We just need some larger equivalence classes of elements. For example, consider the divisibility relation on $\mathbb{Z}$, or on the polynomial ring $F[x]$. It's clearly not symmetric, but it's also not antisymmetric; $n$ and $-n$ divide each other in $\mathbb{Z}$, and any two constant multiples of the same polynomial divide each other in $F[x]$.