Question

If r is a relation in n*n defined by (a,b) r(c,d) if and only if a+d= c+ b show that r is an equivalance relation

Answer 1

Answer:

R is an equivalance relation

Step-by-step explanation:

If r is a relation in n*n defined by (a,b) r(c,d) if and only if a+d= c+ b show that r is an equivalance relation

In N*N defined by

(a,b)R(c,d) if and only if a+d=b+c

Reflexive:

For each (a,b)∈ N

$a+b=b+a$

$\implies\:(a,b)R(a,b)$

$\therefore\:\text{R is reflexive}$

Smmetric:

Let (a,b)R(c,d)

$\implies\:a+d=b+c$

since addition is commutative

$\implies\:c+b=d+a$

$\implies\:(c,d)R(a,b)$

$\therefore\:\text{R is symmetric}$

Transitive:

Let (a,b)R(c,d) and (c,d)R(e,f)

$\implies\:$ a+d=b+c and c+f=d+e

$\implies\:$ a+d=b+c....(1) and d+e=c+f........(2)

(1)-(2)==> a-e=b-f

==> a+f=b+e

(a,b)R(e,f)

$\therefore\:\text{R is transitive}$

Hence R is an equivalance relation