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Question

Let R be the relation over the set N×N and is defined by a,bRc,da+d=b+c, Then R is


A

Reflexive only

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B

Symmetric only

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C

Transitive only

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D

An equivalence relation

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Solution

The correct option is D

An equivalence relation


Explanation for correct option:

Checking whether R is reflexive, symmetric, transitive or an equivalence relation

Step 1: Given information

R is Relation in N×N

N is a set of natural number

a,bRc,da+d=b+c

We know that, if a relation is Reflexive, Symmetric, and Transitive then we call it an Equivalence relation.

Step 2: Checking For Reflexive relation

Here, a,bRa,b because

a+b=a+b which is true in N×N

Therefore, R is reflexive

Step 3: Checking For Symmetric relation

a,bRc,da+d=b+cc+b=d+ac,dRa,b

Therefore, R is symmetric

Step 4: Checking For Transitivity relation

let a,bRc,d and c,dRe,f

This gives

a+d=b+cand c+f=d+e

Adding both the equation above, we get:

a+d+c+f=b+c+d+ea+f=b+ea,bRe,f

Therefore, R is Transitive

Step 5: Checking for an Equivalence relation

Since, R is Reflexive, Symmetric, and Transitive, it is an Equivalence relation.

Hence, option (D) is the correct answer.


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