# Working with partial order relations in discrete math.

Let S = {0,1} and consider the partial order relation R defined on S X S X S as follows: for all ordered triples (a, b, c) and (d, e, f) in S X S X S.

( a, b, c ) R ( d, e, f ) <-> a ≤ d, b ≤ e, c ≤ f,

where ≤ denotes the usual "less than or equal to" relation for real numbers. Do the maximal, greatest, minimal and least elements exist? If so, which are they?

#### Solution Preview

... the greatest element. We know a maximal element is the greatest one in a partial order chain. This element, say (a,b,c) has to be (1,1,1). If not, then a<1 or b<1 or c<1. In each case we have (a,b,c)R(1,1,1) and ...