Question

Let R be a relation over the set N × 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

Answer: (D)

An equivalence relation

Answer 2

We have $( a , b ) R ( a , b )$for all $( a , b ) \in N \times N$

Since $a + b = b + a$. Hence, R is reflexive.

R is symmetric for we have$( a , b ) R ( c , d )$⇒ $a + d = b + c$

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

Then by definition of R, we have

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

whence by addition, we get

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

Hence,$( a , b ) R ( e , f )$

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