Explain why this relation has a reflexive, symmetric, antisymmetric, and transitive propery

Question

Let S = {1, 2, 3} Let R = {(1,1),(3,3),(2,2)}

So the answer is that it is reflexive, symmetric, antisymmetric, and transitive. I understand that it is reflexive, however I do not understand how it could be symmetric, antisymmetric, and transitive.

Can anyone explain why it would have the 3 other properties?

Thank you.

Answer 1

$R$ is symmetric if $\langle x,y\rangle\in R$ implies that $\langle y,x\rangle\in R$. Pick any member of $R$ to be $\langle x,y\rangle$, say $\langle 2,2\rangle$; then $x=2$ and $y=2$, so $\langle y,x\rangle=\langle 2,2\rangle$, and the reversed pair is identical to the original one and is therefore in $R$ as well. Since all of the pairs in $R$ have both components equal, the same thing happens with each of them: the reversed pair is still in $R$, because it’s the same as the original pair. Thus, $R$ is symmetric.

$R$ is antisymmetric if the following is true: whenever $\langle x,y\rangle$ and $\langle y,x\rangle$ both belong to $R$, then $x=y$. For what values of $x$ and $y$ is it true that $\langle x,y\rangle$ and $\langle y,x\rangle$ both belong to this relation $R$? The only time we have $\langle x,y\rangle\in R$ is when $x=y$; in that case certainly both $\langle x,y\rangle$ and $\langle y,x\rangle$ are in $R$, since they’re the same pair, and it is indeed true that $x=y$, as required for antisymmetry. Thus, $R$ is antisymmetric.

Transitivity is similar. Transitivity of $R$ requires that whenever $\langle x,y\rangle$ and $\langle y,z\rangle$ both belong to $R$, then so does $\langle x,z\rangle$. For this particular relation $R$ the only time $\langle x,y\rangle$ belongs to $R$ is when $x=y$, and the only time $\langle y,z\rangle$ belongs to $R$ is when $y=z$, so the only time $\langle x,y\rangle$ and $\langle y,z\rangle$ both belong to $R$ is when $x=y=z$. But in that case $\langle x,z\rangle\in R$, so the requirement for transitivity is met.