Define a relation R on R, the set of all real numbers as follows: For every x , y ∈ R ,
x R y   ⇔ x 2 ≤ y 2 .

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 a relation R on the real number set ℝ as follows.

xRy⇔x2≤y2. For all x,y∈ℝ.

Concept used:

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

Calculation:

Objective is to identify whether the relation is partial order relation or not.

A relation R on the set A is a partial order relation if R is reflexive, antisymmetric, and transitive.

A relation R on the set A is antisymmetric if for all a and b in A, if aRb and bRa then a=b.

Here, the relation R defined on a real number set xRy⇔x2≤y2 is not a partial order relation since R is not antisymmetric.

Counterexample