Question

If we define a relation R on the set $N\times N$ as $\left( a,b \right)R\left( c,d \right)\Leftrightarrow a+d=b+c$ for all $\left( a,b \right),\left( c,d \right)\in N\times N$, then the relation is,
A. Symmetric only
B. Symmetric and transitive only
C. Equivalence relation
D. Reflexive only

Answer 1

For solving this type of questions you should know about the symmetric property, transitive property and reflexive property and the equivalence of the relation. So, according to the reflexive property $a\sim a$, it is always valid for this. The symmetric property states that if $a\sim b$, then $b\sim a$. And the transitive property is defined as, if $a\sim b$ and $b\sim c$, then $a\sim c$. And all these are valid for the sets. If all these three are valid, then the relation will be an equivalence relation.

Complete step by step answer:
So, in the question it is given that, $\left( a,b \right)R\left( c,d \right)\Leftrightarrow a+d=b+c$.
So, if we take $\left( a,a \right)R\left( a,a \right)\Leftrightarrow a+a=a+a$. By the property of reflexive, we can say that here R is reflexive.
Now according to our question, if we apply the symmetric property, then let us consider,
$\begin{aligned} & \left( a,b \right)R\left( c,d \right)\Leftrightarrow a+d=b+c \\\ & \Rightarrow c+b=d+a\Leftrightarrow \left( c,d \right)R\left( a,b \right) \\\ \end{aligned}$
So, here we can say that it is clear that R is symmetric. And this is the step to check the symmetricity of R.
Now if we check R by applying transitive property, then let us consider,
$\left( a,b \right)R\left( c,d \right)$ and $\left( c,d \right)R\left( e,f \right)$
$\Rightarrow a+d=b+c$ and $c+f=d+e$
If we merge both with one another with same sides, then we have,
$a+d+c+f=b+c+d+e$
So, here, $a+f=b+e\Rightarrow \left( a,b \right)R\left( e,f \right)$
By the property of transitive, it is clear that this is transitive also. So, we can say that all three properties transitive, reflexive and symmetric are completely satisfied by it. So, R is an equivalence relation.

So, the correct answer is “Option C”.

Note: When you solve this question then you should apply all the properties in its proper way because they are the only way to clear this type of questions and R will be equivalence when all three properties are completely satisfied and if any one or two or all properties do not satisfy, then it is not equivalence.