Antisymmetric relation

Definition

Let $R$ be a binary relation on a set $A$. The relation $R$ is antisymmetric when, for all $a, b \in A$, $a \mathrel{R} b$ and $b \mathrel{R} a$ imply $a = b$.

Remarks

The relation $\le$ on $\mathbb{R}$ and the relation $\subseteq$ on sets are antisymmetric. Divisibility is antisymmetric on $\mathbb{N}$ but not on $\mathbb{Z}$, because $1 \mid -1$ and $-1 \mid 1$. Antisymmetric is not the opposite of symmetric: equality is both symmetric and antisymmetric.

Explanations

L1 human

If $a \le b$ and $b \le a$, then $a = b$. So for $\le$, a two-way relation between two elements forces them to be the same.