Question

The following relation is defined on the set of real numbers:

$aRb$, if $a-b>0$
Find whether the relation is reflexive, symmetric or transitive.

Answer 1

Given the relation is defined on the set of real numbers:

$aRb$ if $a-b>0$.

Now this relation is not reflexive as ${}_a{\mathrm{R/}}_a$, since $a-a=0\ngtr 0$ $\forall a\in R$.

Similarly the relation is not symmetric as ${}_a{R}_b{\nRightarrow }_b{R}_a$ since $a-b>0$ this gives $b-a<0$ $\forall a,b\in R$.

But the relation is transitive as ${}_a{R}_b$ and ${}_b{R}_c{\Rightarrow }_a{R}_c$ since $a-b>0$ and $b-c>0$ simply gives $(a-b)+(b-c)>0$ or $a-c>0$ $\forall a,b,c\in R$.