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  • Provide solution for RD Sharma maths Class 12 Chapter Relations Exercise 1.1 Question 2 maths textbook solution.

Provide solution for RD Sharma maths Class 12 Chapter Relations Exercise 1.1 Question 2 maths textbook solution.

Answers (1)

Answer:

R_{1} is reflexive and transitive but not symmetric

R_{2} is reflexive, symmetric, and transitive.

R_{3} is transitive but neither reflexive nor symmetric

R_{4} is neither reflexive nor symmetric nor transitive

Hint :

 A relation R on set A is

 Reflexive relation:

If  (a, a) \in R for every a \in A

Symmetric relation:

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

Transitive relation:

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

Given :

Set  A=\{a, b, c\}

\begin{aligned} &R_{1}=\{(a, a),(a, b),(a, c),(b, b),(b, c),(c, a),(c, b),(c, c)\} \\ &R_{2}=\{(a, a)\} \\ &R_{3}=\{(b, c)\} \\ &R_{4}=\{(a, b),(b, c),(c, a)\} \end{aligned}

Consider   R_{1}=\{(a, a),(a, b),(a, c),(b, b),(b, c),(c, a),(c, b),(c, c)\}

Reflexive :

Given (a, a),(b, b) and (c, c) \in R_{1}

So, R_{1} is reflexive.

For Symmetric:

We see that the ordered pairs obtained by interchanging the components of R_{1} are not in R_{1}.

For ex : (a, b) \in R_{1} \text { but }(b, a) \notin R_{1}

So, R_{1} is not symmetric.

For Transitive:

Here, (a, b) \in R_{1} \text { and }(b, c) \in R_{1} but (a, c) \in R_{1}

So, R_{1} is transitive

(ii) Consider R_{2}

R_{2}=\{(a, a)\}

Reflexive:

clearly (a, a) \in R_{2}

 So, R_{2} is reflexive.

Symmetric:

Clearly (a,a)\in \; R_{2}

 So, R_{2}  is symmetric.

Transitive:

R_{2} is a transitive relation, since there is only one element in it.

(iii)  Consider R_{3}

R_{3}=\{(b, c)\}

Reflexive:

Here neither (b, b) \notin R_{3} nor (c, c) \notin R_{3}

So, R_{3}  is not reflexive

Symmetric:

Here neither (b, c) \in R_{3} nor (c, b) \notin R_{3}

So, R_{3} is not symmetric.

Transitive:

R_{3} has only one element

Hence R_{3} is transitive.

(iv)   Consider  R_{4}=\{(a, b),(b, c),(c, a)\}

 Reflexive:

Here  (a, b) \in R_{4} \text { but }(b, a) \notin R_{4}

So, R_{4} is not reflexive

Symmetric:

Here (a, b) \in R_{4} \text { but }(b, a) \notin R_{4}

So,R_{4} is not symmetric

Transitive:

Here (a, b) \in R_{4},(b, c) \in R_{4} but  (a, c) \notin R_{4}

Hence R_{4} is not transitive.

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