Describing relations

Question

(a). Describe all relations $R$ on $A$ which are simultaneously symmetric and antisymmetric.

(b). Describe all relations $R$ on $A$ which are reflexive, symmetric, and antisymmetric.

I have no idea what this even means. I know what symmetric, antisymmetric, reflexive and simultaneous means but I don't know what it means for how general this question this.

Thanks for the help!

Answer 1

HINT: Suppose that $R$ is a symmetric relation on $A$; then whenever $\langle x,y\rangle\in R$, we also have $\langle y,x\rangle\in R$. Informally, for any $x,y\in A$, either both $\langle x,y\rangle$ and $\langle y,x\rangle$ are in $R$, or neither is in $R$: $R$ cannot contain just one of the two.

Now suppose that $R$ is an antisymmetric relation on $A$; then whenever $R$ contains both $\langle x,y\rangle$ and $\langle y,x\rangle$, it turns out that $x=y$. In other words, if $x,y\in A$ and $x\ne y$, then $R$ cannot contain both $\langle x,y\rangle$ and $\langle y,x\rangle$.

Finally, suppose that $R$ is both symmetric and antisymmetric. If $x,y\in A$ and $x\ne y$, then $R$ cannot contain both $\langle x,y\rangle$ and $\langle y,x\rangle$, because $R$ is antisymmetric; and because $R$ is symmetric, it must contain either both of them or neither of them. So $R$ must contain ... ?

Note that this says nothing about whether $R$ contains pairs like $\langle x,x\rangle$; those are considered in the second part of the question.

Answer 2

The second question is actually simpler to answer, so let's start by it.

(b). If $x R y$, then $yRx$ by symmetry. But antisymmetry means that $x=y$. Conversely, if $x=y$, then $xRy$ since $R$ is reflexive. Thus, we proved that $xRy\ \Leftrightarrow x=y$, i.e. $R$ is the equality. To sum up, there is exacyly one symmetric, antisymmetric, reflexive relation on $A$, the equality.

(a) Similarly, if $xRy$, then $x=y$, but the converse is false, since $R$ is not assumed to be reflexive. Then $R$ is completely determined by the subset $B$ of $A$ of elements $x$ such that $xRx$.

Alternative ways of saying this is: the set of all symmetric and antisymmetric relations on $A$ is in bijection with the power set $\mathcal{P}(A)$. Or, since a relation is just a subset of $A\times A$, a relation $R$ is symmetric and antisymmetric if and only if it is included in the diagonal $\Delta=\{ (x,x)\ |\ x\in A \}$.