Let P be the set of all people who have ever lived and define a relation R on P as follows: For every r ,   s ∈ P , is an ancestor of s or r = s .
Is R a partial order relation? Prove or give a counterexample.

To prove or give the counter example for R to be a partial order relation.

Given information:

Let P be the set of all people who have ever lived and define a relation R on P as follows: for all r,s∈P, rRs⇔ r is an ancestor of s or r=s.

Calculation:

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

So let us verify the following all three properties for R to be a partial order relation.

Reflexivity- Suppose r∈P⇒rRr because r=r. Therefore the relation R is reflexive.

Antisymmetry- Suppose r,s∈P, such that rRs⇔ r is an ancestor of s or r=s   ....(1).

And sRr⇔ s is an ancestor of r or s=r    ....(2)