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Define partial order and total order of relations
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A relation T on a set M is called a partial order relation when it satisfies the following properties:
- It's reflexive: (x, x) is in T for every x in M.
- It's antisymmetric: If (x, y) is in T and (y, x) is in T, then x = y.
- It's transitive: If (x, y) is in T and (y, z) is in T, then (x, z) is in T.
A binary relation R on a non-empty set A is a total order if the relation is
- connex
- antisymmetric, and
- transitive.