Let $S=\{1,2,3\}$ and $R$ be a symmetric but not reflexive relation over $S$. Because $R$ is a set of all relations (under the given conditions) over $S$ then $\subseteq$ is a partial order over $R$. Prove that there's the least element in $R$ and prove that $R$ doesn't have the greatest element.
I think that the empty set is the least element in $R$ because for empty set is in any subset of $R$ by definition of empty set. Therefore the empty set is in relation with every element of $R$.
I'm confused why $R$ doesn't have the greatest element. Isn't it:
$$
X=\{(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\}
$$
?
$X$ contains all the symmetric pairs in $R$ and it contains every other element of $R$.
Best Answer
Let's first create a not confusing question.
Let S = {1, 2, 3} and
K = { R subset S×S : R is symmetric and not reflexive }.
Give K the subset order. Does K have a maximum?
A = { (1,1) } in K. B = { (2,2) } in K. C = { (3,3) } in K.
If K had a maximum M, then { (1,1), (2,2), (3,3) } subset M.
As M is reflexive, a contradiction ensues.