Question

Let $S$ be the set of all real numbers and let $R$ be a relation in $S$ , defined by $R=\left\{ \left( a,b \right):a\le b \right\}$ . Show that $R$ is reflexive and transitive but not symmetric.

Answer 1

Hint:For solving this question we will consider some suitable examples for proving that relation $R=\left\{ \left( a,b \right):a\le b \right\}$ is reflexive and transitive but not symmetric comfortably.

Complete step-by-step answer:
Given:
It is given that there is a set $S$ of all real numbers and let $R$ be a relation in $S$ , defined by $R=\left\{ \left( a,b \right):a\le b \right\}$ and we have to prove that $R$ is reflexive and transitive but not symmetric.
Now, we will prove that the given relation is reflexive, transitive but not symmetric one by one.
1. If $a=b$ where $a$ is any real number then it is always true that $a=b$ so, $a\le b$ holds. Now, we conclude that $\left( a,a \right)\in R$ where $a$ is any real number from the set $S$ . Thus, the given relation $R$ is reflexive.

2. If $a$ and $b$ are two different real numbers such that $a\le b$ . Then, it is not correct to write $b\le a$ and we can see this with the help of an example: take $a=2$ , $b=3$ then, it is evident that $a\le b$ but, $b\le a$ doesn’t hold. Now, we can conclude that if $\left( a,b \right)\in R$ then we cannot always say that $\left( b,a \right)\in R$ always. Thus, if $\left( a,b \right)\in R$ then $\left( b,a \right)\notin R$ so, the given relation is not symmetric.

3. If $a$ , $b$ and $c$ are three different real numbers such that $a\le b$ and $b\le c$ . Then, it is always necessary that $a\le c$ and we can see this with the help of an example: take $a=2$ , $b=3$ and $c=4$ then, it is evident that $a\le b$ , $b\le c$ and automatically $a\le c$ . Now, we conclude that if $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$ then, it is always necessary that $\left( a,c \right)\in R$ . Thus, if $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$ then $\left( a,c \right)\in R$ so, the given relation is transitive.
Now, as we have proved that the given relation is reflexive, transitive but not symmetric.
Hence, proved.

Note: Here, the student must try to understand the relation given in the problem then prove that the relation is reflexive, transitive but not symmetric separately and don’t mix up their conditions with each other to avoid the confusion. Moreover, we should understand with the help of suitable examples for better clarity of the concept.