Question

If \[\mathbb{Z}\] is the set of integers. Then, the relation \[R=\left\{ \left( a,b \right):1+ab>0 \right\}\] on \[\mathbb{Z}\] is
(a) Reflexive and transitive but not symmetric
(b) Symmetric and transitive but not reflexive
(c) Reflexive and symmetric but not transitive
(d) An equivalence relation

Answer 1

Hint: We solve this problem by checking the given relation whether satisfies reflexive, symmetric, and antisymmetric relations.
We use the notation of relation that if A related to B in relation R as \[ARB\]
(1) A relation R is said to be reflexive if \[ARA\] satisfies the given relation.
(2) If \[ARB\] satisfies the relation and \[BRA\] satisfies the relation then that relation is said to be symmetric
(3) If \[ARB\] and \[BRC\] satisfies the relation then \[ARC\] also satisfies the relation then that relation is said to be transitive.

Complete step by step answer:
We are given that the relation is defined as
\[R=\left\{ \left( a,b \right):1+ab>0 \right\}\]
Now, let us check for the reflexive relation.
(1) A relation R is said to be reflexive if \[ARA\] satisfies the given relation.
Let us take the element in the relation R as \[\left( a,a \right)\]
By substituting the element in the given condition we get
\[\begin{align}
  & \Rightarrow 1+\left( a\times a \right)>0 \\
 & \Rightarrow {{a}^{2}}>-1 \\
\end{align}\]
Here we can see that the above equation satisfies for all \[a\in \mathbb{Z}\]
So, we can conclude that the given relation is reflexive.
Now, let us check for symmetric relations.
(2) If \[ARB\] satisfies the relation and \[BRA\] satisfies the relation then that relation is said to be symmetric
Let us assume that the element \[\left( a,b \right)\] belongs to given relation R then by using the condition of relation we get
\[\begin{align}
  & \Rightarrow 1+ab>0 \\
 & \Rightarrow 1+ba>0 \\
\end{align}\]
Here we can see that the element \[\left( b,a \right)\] also satisfies for all \[a,b\in \mathbb{Z}\]
Therefore, we can conclude that the given relation is symmetric.
Now, let us check for transitive relation
(3) If \[ARB\] and \[BRC\] satisfies the relation then \[ARC\] also satisfies the relation then that relation is said to be transitive.
Let us assume that the elements \[\left( a,b \right),\left( b,c \right)\] belongs to given relation then by using the condition we get
\[\begin{align}
  & \Rightarrow 1+ab>0 \\
 & \Rightarrow 1+bc>0 \\
\end{align}\]
Now, by adding the above two equation we get
\[\Rightarrow 2+b\left( a+c \right)>0\]
Here we can see that there is no further calculation to prove that \[1+ac>0\]
This means that the element \[\left( a,c \right)\] may or may not belong to a given relation R.
But we know that in order to call it a relation it should satisfy for all values
Therefore we can conclude that a given relation is not transitive.
Hence the given relation is reflexive and symmetric but not transitive.
So, option (c) is the correct answer.

Note:
We need to note that in the checking of transitive relation we need to prove the condition that
\[\Rightarrow 1+ac>0\]
Here, we may check by using the examples, but we cannot check for all the integers where it satisfies or not.
If we are able to conclude that it satisfies for all \[a c\in \mathbb{Z}\] then we can conclude that this relation is transitive. In the same way, if we cannot confirm perfectly then we can give that it is not transitive.