Question

Let $R$ be the relation over the set $N\times N$ and is defined by $\left(a,b\right)R\left(c,d\right)\Rightarrow a+d=b+c$, Then $R$ is

• A. Reflexive only
• B. Symmetric only
• C. Transitive only
• D. An equivalence relation

Answer 1

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 $N\times N$

$N$ is a set of natural number

$\left(a,b\right)R\left(c,d\right)\Rightarrow a+d=b+c$

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, $\left(a,b\right)R\left(a,b\right)$ because

$a+b=a+b$ which is true in $N\times N$

Therefore, $R$ is reflexive

Step 3: Checking For Symmetric relation

$$\begin{gathered} \left(a,b\right)R\left(c,d\right) \\ \Rightarrow a+d=b+c \\ \Rightarrow c+b=d+a \\ \Rightarrow \left(c,d\right)R\left(a,b\right) \end{gathered}$$

Therefore, $R$ is symmetric

Step 4: Checking For Transitivity relation

let $\left(a,b\right)R\left(c,d\right)$ and $\left(c,d\right)R\left(e,f\right)$

This gives

$a+d=b+c$and $c+f=d+e$

Adding both the equation above, we get:

$$\begin{gathered} a+d+c+f=b+c+d+e \\ \Rightarrow a+f=b+e \\ \Rightarrow \left(a,b\right)R\left(e,f\right) \end{gathered}$$

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.