Question

Let $R$ be a relation defined in the set of real numbers by $aRb\iff 1+ab>0$. Then $R$ is

• A. Equivalence relation
• B. Transitive
• C. Symmetric
• D. Anti-symmetric

Answer 1

The correct option is C Symmetric

i) Reflexive: Let R' be the set of real numbers
Let $a\in R^{\prime }\Rightarrow 1+a.a=1+a^2>0$
$\Rightarrow (a,a)\in R$ i.e R is reflexive

ii) Symmetric: $a,b\in R^{\prime }$
Let $(a,b)\in R\Rightarrow 1+a.b>0\Rightarrow 1+b.a>0\Rightarrow (b,a)\in R$
i.e R is symmetric

iii) Transitive: The relation R is not transitive because we find that $(2,\frac{1}{2})\in R$ and $(\frac{1}{2},-2)\in R$
but $(2,-2)\notin R$
since $1+2(-2)=-3$ which is not positive
Hence R is reflexive, symmetric but not transitive