Equivalence relations proof example?

Question

Let $A$ = {$a,b,c$}. Give an example of a relation on $A$ that is anti-symmetric, reflexive on $A$ and symmetric.

The first thing that one must do to proceed with this question is to first define what these terms mean.

Reflexive : R is reflexive on $A$ iff for all $x∈ A$, $xRx.$

Symmetric : R is symmetric iff for all $x$ and $y∈ A$ if $xRy,$ then $yRx.$

Anti-symmetric : iff for all $x,y∈ A,$ if $xRy$ and $yRx$ then $x=y.$

Would this be the right example for it being reflexive? $\Rightarrow$ {$(a,a),(b,b),(c,c)$}

Would symmetric be like this {$(a,b),(b,a),(c,a)$}.

Am I on the right track?

Answer 1

You have reflexive right.   That is the minimal reflexive relation for the set.   You can include others, but you must have at least these three pairs.

A symmetric relation needs to have no exceptions.   You can't include just $(c,a)$.   If any pair exists in the relation, so too must its symmetric pair.

Other than that, you are on the right trail.

Now go on to antisymetric. ...