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

Answer 1

The correct option is A Neither a Partial Order nor an Equivalence Relation

$(x,y)R(u,v)$ iff $x<u$ and $y>v$
$(x,x)\text{/}R(x,x)$ since $x\text{/}<x$ and $x\text{/}>x$
So $R$ is not reflexive.
$\therefore R$ is neither a partial order, nor an equivalent relation.