Define a relation R on Z, the set of all integers as follows: For every m , n ∈ Z ,
m R n ⇔ m + n   is even .

Is R a partial order relation? Prove or give a counterexample.

To check:

Whether R is a partial order relation or not.

Given information:

Consider the relation R on the set of integers ℤ,

For all m,n∈ℤ, mRn⇔m+n is even.

Concept used:

A relation that is reflexive, antisymmetric, and transitive is called a partial order.

Calculation:

Objective is to identify whether R is a partial order relation or not

If R is reflexive, antisymmetric and transitive, then R is called a partial order relation on a set A.

For all a,b∈A, a relation R is antisymmetric if aRb and bRa then a=b