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

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

**Answer:** **(A)** **Explanation:** An **equivalence relation** on a set x is a subset of x*x, i.e., 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.

An **partial order**** relation** on a set x is a subset of x*x, i.e., 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.

An **total order**** relation** a set x is a subset of x*x, i.e., 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.

As given in question, a relation R is defined on ordered pairs of integers as follows: (x,y) R(u,v) if x < u and y > v , reflexive property is not satisfied here, because there is > or < relationship between (x ,y) pair set and (u,v) pair set . Other way , if there would have been x <= u and y>= v (or x=u and y=v) kind of relation among elements of sets then reflexive property could have been satisfied. Since reflexive property in not satisfied here , so given realtion can not be **equivalence, partial order**or total order relation.

So, option (A) is correct.

This solution is contributed by **N****irmal Bharadwaj.**

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