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+7 votes

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
asked in Set Theory & Algebra by Loyal (4.1k points)  
edited by | 400 views

3 Answers

+11 votes
Best answer
ans is (A).. because the relation is not reflexive.. which is a necessary condition for both partial order and equivalence realtion..!!
answered by Loyal (4.8k points)  
selected by
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.
+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 (1.7k points)  
0 votes

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 Active (2k points)  

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