TITLE: Is my understanding of antisymmetric and symmetric relations correct?
QUESTION [19 upvotes]: So I'm having a hard time grasping how a relation can be both antisymmetric and symmetric, or neither. 
Are my examples correct?
symmetric & antisymmetric 
R ={(1,1),(2,2),(3,3)}

not symmetric & not antisymmetric
R = { (1,2),(2,1),(3,4) }

REPLY [27 votes]: Here’s a way to think about symmetry and antisymmetry that some people find helpful. A relation $R$ on a set $A$ has a directed graph (or digraph) $G_R$: the vertices of $G_R$ are the elements of $A$, and for any $a,b\in A$ there is an edge in $G_R$ from $a$ to $b$ if and only if $\langle a,b\rangle\in R$. Think of the edges of $G_R$ as streets. The properties of symmetry, antisymmetry, and reflexivity have very simple interpretations in these terms:

$R$ is reflexive if and only if there is a loop at every vertex. (A loop is an edge from some vertex to itself.)
$R$ is symmetric if and only if every edge in $G_R$ is a two-way street or a loop. Equivalently, $G_R$ has no one-way streets between distinct vertices.
$R$ is antisymmetric if and only every edge of $G_R$ is either a one-way street or a loop. Equivalently, $G_R$ has no two-way streets between distinct vertices.

This makes it clear that if $G_R$ has only loops, $R$ is both symmetric and antisymmetric: $R$ is symmetric because $G_R$ has no one-way streets between distinct vertices, and $R$ is antisymmetric because $G_R$ has no two-way streets between distinct vertices.
To make a relation that is neither symmetric nor antisymmetric, just find a digraph that has both a one-way street and a two-way street, like this one:
$$0\longrightarrow 1\longleftrightarrow 2$$
It corresponds to the relation $R=\{\langle 0,1\rangle,\langle 1,2\rangle,\langle 2,1\rangle\}$. on $A=\{0,1,2\}$.