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Provide solution for RD Sharma maths Class 12 Chapter Relations Exercise 1.1 Question 3 Subquestion (ii) maths textbook solution.

Answers (1)

Answer:  R_{2} is reflexive and symmetric but not 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, c \in A


R_{2} \text { on } Z defined by (a, b) \in R_{2} \Leftrightarrow|a-b| \leq 5

Solution :


Let a be an arbitrary element of R_{2}

Then, a \in R_{2}

On applying the given condition, we get

|a-a|=0 \leq 5

So, R_{2} is reflexive

Symmetry :

Let (a, b) \in R_{2} \quad|a-b| \leq 5

[Since  |a-b|=|b-a| ]

Then |b-a| \leq 5

(b, a) \in R_{2}

So, R_{2} is symmetric.

Transitivity :

Let   (1,3) \in R_{2} \text { and }(3,7) \in R_{2}

|1-3| \leq 5 \text { and }|3-7| \leq 5

But,    |1-7| \leq 5

(1,7) \neq R_{2}

So, R_{2} is not transitive.

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