9.6 Partial Orderings

Question 1

Partial Order:

POSET?

Answer 1

A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set S together with a partial ordering R is called a partially ordered
set, or poset, and is denoted by (S, R). Members of S are called elements of the poset.

Recall theory of origin (why?)

Question 2

denote that (a, b) ∈ R in an arbitrary
poset (S, R).

Answer 2

Customarily, the notation a b is used to denote that (a, b) ∈ R in an arbitrary
poset (S, R). This notation is used because the less than or equal to relation on the set of real
numbers is the most familiar example of a partial ordering and the symbol is similar to the
≤ symbol. (Note that the symbol is used to denote the relation in any poset, not just the less
than or equal to relation.) The notation a ≺ b denotes that a b, but a ≠ b. Also, we say “a is
less than b” or “b is greater than a” if a ≺ b.