Question

Consider the relation Ron A = {1, 2, 3}, where R= {(2,2),(1,3).(3, 2)}. List the pairs that need to be included in R.to make it a partial order,

Answer 1

A relation R on A is called a Partial Order if and only if R is reflexive, antisymmetric and transitive.

A relation R is said to be reflexive ∀ x ∈ A then (x,x) ∈ R

A relation R is said to be antisymmetric ∀(x,y)∈R then (y,x) does not belong to R

A relation R is said to be transitive ∀(x,y)∈R and (y,z)∈R then (x,z) ∈ R

Given A={1,2,3}

Relation R={(2,2),(1,3),(3,2)}

For R to be reflexive we need to add pairs (1,1) and (3,3)

R={(1,1),(2,2),(3,3),(1,3),(3,2)}

Now R is a reflexive relation

(1,3) is in R but there is no pair (3,1).

(3,2) in R but there is no pair(2,3).

Hence R is a antisymmetric relation

(1,3) and (3,2) ∈ R so we need to add (1,2) to R to make relation transitive

R={(1,1),(2,2),(3,3).(1,3),(3,2),(1,2)}

So now R is reflexive, antisymmetric and transitive. Hence R is partial order relation

We added pairs (1,1), (3,3), (1,2) to the relation R to make R partial order.