Question

The following relation is defined on the set of real numbers: aRb if \[1+ab>0\]. Find whether the relation is reflexive, symmetric or transitive.

Answer 1

Hint: We will use the definitions of reflexive, symmetric and transitive relations to solve this question. A relation is a reflexive relation If every element of set A maps to itself. A relation in a set A is a symmetric relation if \[({{a}_{1}},{{a}_{2}})\in R\] implies that \[({{a}_{2}},{{a}_{1}})\in R\], for all \[{{a}_{1}},{{a}_{2}}\in A\]. A relation in a set A is a transitive relation if \[({{a}_{1}},{{a}_{2}})\in R\] and \[({{a}_{2}},{{a}_{1}})\in R\] implies that \[({{a}_{1}},{{a}_{3}})\in R\] for all \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\in A\].

Complete step-by-step answer:
Before proceeding with the question we should know about the concept of relations and different types of relations that are reflexive, symmetric and transitive relations.
A relation in set A is a subset of \[A\times A\]. Thus, \[A\times A\] is two extreme relations.
A relation in a set A is a reflexive relation if \[(a,a)\in R\], for every \[a\in A\].
A relation in a set A is a symmetric relation if \[({{a}_{1}},{{a}_{2}})\in R\] implies that \[({{a}_{2}},{{a}_{1}})\in R\], for all \[{{a}_{1}},{{a}_{2}}\in A\].
A relation in a set A is a transitive relation if \[({{a}_{1}},{{a}_{2}})\in R\] and \[({{a}_{2}},{{a}_{1}})\in R\] implies that \[({{a}_{1}},{{a}_{3}})\in R\] for all \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\in A\].
A relation in a set A is an equivalence relation if R is reflexive, symmetric and transitive.
We will first check reflexivity. Now let a be an arbitrary element of R. Then,
\[\begin{align}
  & \Rightarrow 1+a\times a>0 \\
 & \Rightarrow 1+{{a}^{2}}>0 \\
\end{align}\]
Now we know that the square of any number is positive. So the given relation is reflexive.
Now moving on to symmetry. Let \[(a,b)\in R\].
\[\begin{align}
  & \Rightarrow 1+ab>0 \\
 & \Rightarrow 1+ba>0 \\
\end{align}\]
This implies that \[(b,a)\in R\]. Hence the given relation is symmetric.
Finally, we will check for transitivity. Let \[(a,b)\in R\] and \[(b,c)\in R\].
\[\Rightarrow 1+ab>0\] and \[\Rightarrow 1+bc>0\] but 1+ac is not greater than 0. This implies that \[(a,c)\notin R\] and hence the given relation is not transitive.
So the given relation is reflexive and symmetric but not transitive.

Note:
We in a hurry can make a mistake in thinking it as a transitive set but we have to check the definition by taking subsets of the given set A. A relation in a set A is a transitive relation if \[({{a}_{1}},{{a}_{2}})\in R\] and \[({{a}_{2}},{{a}_{1}})\in R\] implies that \[({{a}_{1}},{{a}_{3}})\in R\] for all \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\in A\].