Answer to Question #213798 in Discrete Mathematics for Aroosha ch

Question

i)                  Which type of relation is shown in below expression

R1 = { (a,b) | a = b } 

Answer 1

The relation $R_1 = \{ (a,b)\ |\ a = b \}$ is binary. Since $(a,a)\in R_1$ for any $a$, this relation is reflexive. If $(a,b)\in R_1,$ then $a=b,$ and hence $(b,a)\in R_1,$ that is the relation is symmetric. If $(a,b)\in R_1$ and $(b,c)\in R_1,$ then $a=b=c,$ and thus the relation is transitive. Therefore, $R_1$ is equivalence relation.