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Need Solution for R. D.Sharma Maths Class 12 Chapter relations Exercise 1.1 Question 14 sub question 5 Maths Textbook Solution.

Answers (1)

Answer: $R=\{(a, b): a<b\}$

Hint:

A relation R on set A is

Reflexive relation:

If $(a, a) \in R$ for every $a \in A$

Symmetric relation:

If $\left ( a,b \right )$ is true then $\left ( b,a\right )$ is also true for every $a, b \in A$

Transitive relation:

If $(a, b) \text { and }(b, c) \in R, \text { then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$

Given:

We have to give the example of a relation which is transitive but neither symmetric nor reflexive.

Solution:

The relation having properties of being transitive but neither symmetric nor reflexive.

Consider a relation R in $R$ defined as:

$R=\{(a, b): a<b\}$

For any $a \in R$ we have $(a, a) \notin R$ , since $a$ can’t be strictly less than $a$ itself.

Infact $a=a$

∴ Relation $R$ is not reflexive.

Now, $(1,2) \in R(\text { as } 1<2)$

But 2 is not less than 1

∴ $(2,1) \notin R$

∴ $R$ is not symmetric

Now let $(a, b),(b, c) \in R$

$\begin{aligned} &a<b \text { and } b<c \\ &a<c \\ &(a, c) \in R \end{aligned}$

$R$ is transitive.

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