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  • Please Solve R.D. Sharma class 12 Chapter relations Exercise 1.1 Question 12 Maths textbook Solution.

Please Solve R.D. Sharma class 12 Chapter relations Exercise 1.1 Question 12  Maths textbook Solution.

Answers (1)

Answer:

The relation having properties of being transitive and reflexive but not symmetric.

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, then (a, c) \in R  for every a, b, c \in A

Given:

 An integer m  is said to be related to another integer n if m is multiple of n

Solution:

Reflexivity:

\begin{aligned} &\text { Let } m \in Z \\ &m=1 . m \\ &(m, m) \in R \end{aligned}

R is reflexive.

Symmetric:

\begin{aligned} &\text {Let }(a, b) \in R \\ &\Rightarrow \quad a=k b \\ &\Rightarrow \quad b=\frac{1}{k} a \quad \text { but } \frac{1}{k} \notin z \text { if } k \in z \\ &\therefore \quad(b, a) \notin R \end{aligned}

R  is not symmetric.

Transitive:

\begin{aligned} &\text { Let }(a, b) \in R \text { and }(b, c) \in R \\ &a=k b \text { and } b=k^{\prime} c \end{aligned}

Then,  \begin{aligned} a=k k^{\prime} c &\left[k k^{\prime} \in Z\right] \\ a=I c &\left[I=k k^{\prime} \in Z\right] \\ (a, c) \in R & \end{aligned}

 R is transitive.

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