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

(a, b) R (a, b) because a + b = b + a. So, r is reflexive.
(a, b)R (c, d) $\Rightarrow$ a+d = b+c $\Rightarrow$ c+b = d+a
$\Rightarrow$ (c,d) R (a,b)
So, R is symmetric.
(a, b) R (c, d) and (c, d) R (e, f)
$\Rightarrow$ a + d = b + c, c + f = d + e
Adding, a + d + c + f = b + c + d +e
$\Rightarrow$ a + f = b + e
$\Rightarrow$ (a, b) R (e, f).
$\therefore$ R is transitive.