Question

Let $R$ be a relation over the set $N\times N$ and it is defined by $(a,b)R(c,d)\Rightarrow a+d=b+c.$ 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 The Correct Option:

Determining the correct option

Checking reflexivity

The given relation,

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

Putting $a=a$

$\begin{array}{rcl}(a,a)R(a,a)& \Rightarrow & a+a=a+a\\ & \Rightarrow & 2a=2a\end{array}$

Thus it is reflexive.

Checking symmetricity

The given relation,

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

Thus, it is symmetric.

Checking Transitivity

$(a,b)R(c,d)\Rightarrow (c,d)R(e,f)$

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

Thus, it is transitive.

Since, $R$ is reflexive, symmetric and transitive.

Therefore, $R$ is an equivalence.

Hence, option (D) is the correct answer.