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

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,
\[\begin{align}
  & \Rightarrow a-b>0 \\
 & \Rightarrow a-a=0 \\
\end{align}\]
Now we know that 0 is not greater than 0. So the given relation is not reflexive.
Now moving on to symmetry. Let \[(a,b)\in R\].
\[\begin{align}
  & \Rightarrow a-b>0 \\
 & \Rightarrow b-a<0 \\
\end{align}\]
This implies that \[(b,a)\notin R\]. Hence the given relation is not symmetric.
Finally, we will check for transitivity. Let \[(a,b)\in R\] and \[(b,c)\in R\].
\[\Rightarrow a-b>0.....(1)\] and \[\Rightarrow b-c>0.......(2)\]
Adding equation (1) and equation (2) we get,
\[\begin{align}
  & \Rightarrow a-b+b-c>0 \\
 & \Rightarrow a-c>0 \\
\end{align}\]
This implies that \[(a,c)\in R\] and hence the given relation is transitive.
So the given relation is not reflexive and not symmetric but it is transitive.

Note: Remembering the definition of relations and the types of relations is the key here. We in a hurry can make a mistake in thinking it as a symmetric set but we have to check the definition by taking subsets of the given set A and then putting it in the relation.