GATE CSE 2006 | Question: 4

Question

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:

• A.    Neither a Partial Order nor an Equivalence Relation
• B.    A Partial Order but not a Total Order
• C.    A total Order
• D.    An Equivalence Relation

Comments

thank you :)

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1)Given relation (x,y)R(u,v) if x<u and y>v

is not reflexive, not symmetric, transitive, Irreflexive, Antisymmetric, Asymmetric. It is not partial order nor equivalence relation. See @AyushUpadhyaya comment below for more info.

 

2)If question was like relation (x,y)R(u,v) if x<=u and y>=v

then it is reflexive,Antisymmetric, transitive. So, Partial order.

But not total order because neither (0,0)R(2,2) nor (2,2)R(0,0).

It is not equivalence relation also.

It is reflexive, not symmetric, transitive, not Irreflexive, Antisymmetric, not Asymmetric.

Please, correct if anything is wrong here.

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rajankakaniya looks good to me

Answer 1

equivalence relation - a collection R of ordered pairs of elements of x, satisfying certain properties. Write “x R y” to mean (x,y) is an element of R, and we say “x is related to y,” then the properties are:

1. Reflexive: a R a for all a Є R,

2. Symmetric: a R b implies that b R a for all a,b Є R

3. Transitive: a R b and b R c imply a R c for all a,b,c Є R.

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partial order-  a collection R of ordered pairs of elements of x, satisfying certain properties. Write “x R y” to mean (x,y) is an element of R, and we say “x is related to y,” then the properties are:

1. Reflexive: a R a for all a Є R,

2. Anti-Symmetric: a R b and b R a implies that for all a,b Є R

3. Transitive: a R b and b R c imply a R c for all a,b,c Є R.

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total order - a collection R of ordered pairs of elements of x, satisfying certain properties. Write “x R y” to mean (x,y) is an element of R, and we say “x is related to y,” then the properties are:

1. Reflexive: a R a for all a Є R,

2. Symmetric: a R b implies that b R a for all a,b Є R

3. Transitive: a R b and b R c imply a R c for all a,b,c Є R.

4. Comparability : either a R b or b R a for all a,b Є R.

 

so correct option is A
So R is not reflexive.
∴R is neither a partial order, nor an equivalent relation.