Is my understanding of antisymmetric and symmetric relations correct?

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 $A=\{0,1,2\}$.

Symmetric means if $(a,b)$ is there then so is $(b,a)$. Antisymmetric means if $(a,b)$ is there then $(b,a)$ isn't there.

Look at your second example. $(3,4)$ is there. Is $(4,3)$ there? Is the relation symmetric? $(1,2)$ and $(2,1)$ are both there. Is the relation antisymmetric?

One definition of antisymmetry is $$ [(a,b) \wedge (b,a)] \Rightarrow a = b. $$ That is, if $a$ is related to $b$ and vice-versa, then $a$ and $b$ are actually the same element.

In your first example, the lefthand side of the definition above is never invoked for distinct $a$ and $b$. For this reason, you might say the relation is vacuously antisymmetric. The argument for its symmetry is similar. In fact, the only way a relation can be both symmetric and antisymmetric is if all its members are of the form $(x,x)$, like in the example you give.

To violate symmetry or antisymmetry, all you need is a single example of its failure, which Gerry Myerson points out in his answer.