JEE Main & Advanced Mathematics Functions Question Bank Self Evaluation Test - Relation and Functions-II

Let R be a relation over the \[N\times N\] 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

Correct Answer: D

Solution :

[d] we have \[(a,b)R(a,b)\] for all \[(a,b)\in N\times N\]

As \[a+b=b+a.\]Hence, R is reflexive

R is symmetric for we have (a, b) R (c, d)

\[\Rightarrow a+d=b+c\] \[\Rightarrow d+a=c+b\]

\[\Rightarrow a+b=b+c\] \[\Rightarrow (c,d)R(e,f)\]

Then, by definition of R, we have

\[a+d=b+c\] and \[c+f=d+e\]

So, by addition, we get

\[a+d+c+f=b+c+d+c\] or \[a+f=b+e\]

Hence, \[(a,b)R(e,f)\]

Thus, \[(a,b)R(c,d)\] and \[(c,d)R(e,f)\]

\[\Rightarrow (a,b)R(e,f)\]