State whether or not each of the following relations defined

State whether or not each of the following relations defined on X = {1,2,3,4} is a partial order. If any of the relations is not a partial order, state why not.

Solution

In order for the set relation is a partial order, it must be reflexive, antisymmetric and transitive

a) (1,1),(2,2),(3,3),(4,4)

The relation is reflexive since (a,a) belongs to R

The relation is antisymmetric since if (a,b) belongs to R then (b.a) doesn\'t belong to R

The relation is transitive

Hence the relation is a partial order

b) The relation is not a partial order since neither it is reflexive since (1,1) and (2,2) doesn\'t belong to R and also it is not antisymmetric since (1,2) and (2,1) belongs to R

Hence relation is not a partial order

c) Relation is reflexive

Relation is anitsummetric since if (2,4) belongs to R, then (4,2) doesn\'t belong to R

Relation is transitive as well since (a,b),(b,c) belongs to R , (a,c) belongs to R

Hence the relation is a partial order