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:
(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. Anti-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 relation can not be equivalence, partial orderor total order relation.
So, option (A) is correct.
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