What can be said about a relation $R=(A,A,R)$ that is refelxive, symmetric and antisymmetric?

Question

What can be said about a relation $R=(A,A,R)$ that is refelxive, symmetric and antisymmetric?

I know the definitions:

• Reflexive: $xRx$ $\forall$ $x \in A$
• Symmetric: if $\forall a,b \in A, aRb \Rightarrow bRa$
• Antisymmetric: $aRb$ and $bRa \Rightarrow a=b$

Can I get some hints?

Answer 1

Note that if $aRb$ or $bRa$, then $a = b$; this follows since either of $aRb$, $bRa$ implies the other via the symmetry condition, and then the antisymmetry condition yields $a = b$. This in turn implies that $R$ is transitive, since $aRb$ and $bRc$ forces $a = b = c$, and $aRa$ by reflexivity gives us $aRc$, so transitivity follows. Thus $R$, being reflexive, symmetric, and transitive, satisfies the definition of an equivalence relation. The equivalence classes are all singletons; thus the assertions "$a = b$" and "$aRb$" are logically equivalent, and $R$ functions for all the world exactly as does $=$.

Answer 2

Hint: Assume that $aRb$ and use the definition of symmetry and then the definiton of anti-symmetry to deduce an equality.