EDB — 224

Exercises

1. [224]Prerequisites:[23X]↺↻,[1Y5]↺↻.

   Given two relations \(a≤ b\) and \(a{\lt} b\) for \(a,b\in A\), show that these are equivalent:

   • \(a≤ b\) is a (possibly partial) order relation and we identify

     \[ a{\lt}b = (a≤ b\land a\neq b)\quad ; \]

   • \(a{\lt} b\) is an irreflexive and transitive relation and \(\forall x,y\in A\) at most one of \(x{\lt}y~ ,~ x=y ~ ,~ y{\lt}x\) holds; and we identify

     \[ a≤ b = (a{\lt} b\lor a= b)\quad . \]

   This latter \(a{\lt} b\) is called strict (partial) order.