Question

show that the relation R={(a,b):a-b)= even integer and a,b€I}in the set I integers is an equivalence relation.​

Answer 1

Answer:

(i) Since a−a=0 and 0 is an even integer

(a,a)∈R

∴ R is reflexive.

(ii) If (a−b) is even, then (b−a) is also even. then, if (a−b)∈R,(b,a)∈R

∴ The relation is symmetric.

(iii) If (a,b)∈R,(b,c)∈R, then (a−b) is even, (b−c) is even, then $$(a-b

+b-c)=a-c$$ is even.

∴ If (a,b)∈R,(b,c)∈R implies (a,c)∈R

∴ R is transitive.

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