Question

The relation “less than” in the set of natural numbers is
A. Only symmetric
B. Only Transitive
C. Only Reflexive
D. Equivalence relation

Answer 1

Hint: We will be using the concepts of functions and relations to solve the problem. We will be using the definitions of reflexive relation, symmetric relations and transitive relations to verify if each relation holds or not and hence deduce the answer.

Complete step-by-step solution -
Now, we have been given a relation and we have to find whether the relation is reflexive, symmetric, transitive or a combination of these.
Now, we know that reflexive relations are those in which every element is mapped to itself i.e. $\left( a,a \right)\in R$ while symmetric relations are those for which if a R b then b R a. Also, holds and transitive are those relations in which if a R b and b R c then a R c must be held.
Now, we know different types of relations and we will check the given relation for these.
Now, we have been given relation “less than” in the set of natural number so we have,
$aRb\Rightarrow a < b;a,b\in N$
So, for the relation to be reflexive we have $aRa$. So, $a < a$ which is not true for all $a\in N$ hence, the relation is not reflexive.
Now, for transitive, we have if,
$\begin{align}
  & aRb\Rightarrow a < b \\
 & Now,\ bRa\Rightarrow b < a \\
\end{align}$
Since, $bRa$ contradicts $aRb$. Hence, the given relation is not symmetric.
Now, for transitive we have if,
$\begin{align}
  & aRb\Rightarrow a < b..........\left( 1 \right) \\
 & bRc\Rightarrow b < c..........\left( 2 \right) \\
\end{align}$
Now, fro (1) and (2) we have,
a < b < c so, a < c
Therefore, if $aRb\ and\ bRc\Rightarrow aRc$. Hence, the given relation is transitive.
The correct option is therefore (B).

Note: To solve these types of questions it is important to note that a R b means that a is related to b by a relation R. Also these types of questions are solved easily by giving examples and counter examples. Also, we have to check the relation for reflexive, symmetric and transitive relation to check it for equivalence relation.