We say a relation on a set A is antisymmetric if and only if

We say a relation on a set A is antisymmetric if and only if for all a,b A, if aRb and bRa then a = b. We say a relation is a partially ordered relation if it is reflexive, antisymmetric and transitive. Define a relation on Z+ by mRn if and only if m | n. Prove that R is a partially ordered relation.

Solution

For the partial ordered relation, it must be reflexive, antisymmetric and transitive

Relation is Reflexive since m divides m

Relation is not symmetric, since if m divides n, then n will not divide m

Relation is transitive, if m|n and n|p, then m|p

The statement is true, since m|n => n = k1m

n|p, p = k2n

p = k2k1m

m divides p, with the factor k1k2

Hence the relation is reflexive antisymmetric and transitive, hence it is partially ordered relation