494 views

A relation $R$ is defined on ordered pairs of integers as follows: $$(x,y)R(u,v) \text{ if } x<u \text{ and } y>v$$ Then R is:

1.    Neither a Partial Order nor an Equivalence Relation
2.    A Partial Order but not a Total Order
3.    A total Order
4.    An Equivalence Relation
edited | 494 views

ans is (A).. because the relation is not reflexive.. which is a necessary condition for both partial order and equivalence realtion..!!
answered by Loyal (5k points) 8 53 81
selected
not reflexive in all cases
For reflexivity, (X,Y) R (X,Y) , which here requires x<x and y<y and this is not possible in this relation.

Just take an eg. of 3 elements. Let set A={0,1,2}

Find out the relation set according to qs.

Relation R ={  ((0,1),(1,0))  ,   ((1,2),(2,1))   ,  ((0,1),(2,0))  ,   ((0,2),(1,0))  ,  ((0,2),(1,1))  ,   ((0,2),(2,1))  , ((1,2),(2,0))  , ((1,2),(2,1)) }

Check properties of relation R :

1.Reflexive    =  NO

2.Symmetric  = NO

3.Transitivity  = YES

4.Antisymmetric = Yes

So according to properties of POR and Equivalence relation it is neither POR nor Equivalence relation.

## The correct answer is (A) Neither a Partial Order nor an Equivalence Relation

answered by Boss (8.8k points) 3 8 12
+1 vote
For a relation to be partial order or equivalence relation it must be reflexive.
i.e. (x,y) is some element of the set then (x,y)R(x,y), but this doesn't satisfy the given condition of x<x, y>y

Option A
answered by Active (1.9k points) 2 28 50