Question

Given the relation R = {(n, m) | n, m ∈ ℤ, n ≥ m}. Which of the following statements about R is correct?

	R is not a partial order because it is not antisymmetric
	R is not a partial order because it is not reflexive
	R is a partial order
	R is not a partial order because it is not transitive

Answer 1

R is a partial order relation. Because it satisfies all the three properties, it is reflexive,antisymmetric and transitive.

Z is the set of integers.

Lets look at all the properties one by one.

Reflexive:

The relation is reflexive because for every  a ∈ Z,  (a, a) ∈ R

Why? Because any integer is also greter than equal to itself. eg. 2>=2 , -1>=-1 , 3>=3 and so on. This is true for all integers in Z.

Antisymmetric:

It is antisymmetric because whenever (m,n) and (n, m) ∈ R, we have m = n.

Why?

(n, m) ∈ R means n>= m

(m, n) ∈ R means m>= n

Only possibility is both are equal, and we already proved this above in the reflexive part that for every a belongs to Z, (a, a) ∈ R

Transitive:

The relation is transitive because whenever (m, n) and (n, o) ∈ R, we have (m, o) ∈ R.

Example:

4>=2 hence (4, 2) ∈ R

and 2>=1 hence (2, 1) ∈ R,

implies (4, 1) ∈ R. because 4>=1

Since the relation is reflexive, antisymetric and transitive,it is partial order relation.