A relation R is defined on ordered pairs of integers as follows: (x,y)R(u,v) if x v. Then R is: A. Neither ...

Question

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

Accepted Answer

A. Neither a Partial Order nor an Equivalence Relation The relation R uses strict inequalities: (x,y)R(u,v) requires x strictly less than u AND y strictly greater than v. The critical test is reflexivity: can (x,y)R(x,y)? This would require x < x which is false. Since R is not reflexive, it can be neither a partial order (which requires reflexivity) nor an equivalence relation (which also requires reflexivity). Both types of relations require that every element relates to itself.

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