Answer to Question #214131 in Discrete Mathematics for Gourav

Question

Show that the relation p = {(a,b) | a -b is an integer} on the set of real numbers R is equivalence relation

Answer 1

(I). Since $a-a=0$ and is an even integer. $\left(a,\:a\right)\in R$ . Therefore, R is reflexive.

(II). If $\left(a-b\right)$ is even then $\left(b-a\right)$ is also even. Hence, $\left(a-b\right)\in R$ and $\left(b,\:a\right)\in R$ . The relation indicate R is symmetric.

(III). If $\left(a,\:b\right)\in R$ , $\left(b,\:c\right)\in R$ , then $\left(a-b\right)$ is even, $\left(b-c\right)$ is even. Hence, $\left(a-b+b-c\right)=\left(a-c\right)$ and $\left(a,\:c\right)\in R$ . This shows that R is transitive.

Since R is reflexive, symmetric and transitive, it is an equivalence relation.