Definition (Antisymmetry) A binary relation R on a set A is said to antisymmetric if, and only if, for every x A and y A, if x R y and y R x, then x = y. Definition: (partial order). A partial order is a pair D = (D, ) where D is a set and is a partial ordering (i.e. a reflexive, antisymmetric, transitive binary relation) on D. As is the custom in the literature, we abuse notation by writing D for D in contexts where ambiguity seems unlikely to result. Let S be the set {1, 2, 3}. How many partial orderings on S are there?