As a bonus today, I want to define a few more kinds of relations.
A preorder is a relation on a set which is reflexive and transitive. We often write a general preorder as and say that precedes or that succeeds . A set equipped with a preorder is called a preordered set. If we also have that for any two elements and there is some element (possibly the same as or ) that succeeds both of them we call the structure a directed set.
A partial order is a preorder which is also antisymmetric: the only way to have both and is for and to be the same element. We call a set with a partial order a partially-ordered set or a “poset”.
Any set gives a partial order on its set of subsets, given by inclusion: if and are subsets of a set , then precedes if is contained in . This has the further nice property that it has a top element, itself, that succeeds every element. It also has a bottom element, the empty subset, that precedes everything. The same sort of construction applies to give the poset of subgroups of any given group. These kinds of partially-ordered sets are very important in logic and set theory, and they’ll come up in more detail later.
Finally, a partial order where for any two elements and we either have or is called a total order. Total orders show up over and over, and they’re nice things to have around. I must admit, though, that as far as I’m concerned they’re pretty boring in and of themselves.