Question

4.Let R be a relation defined on the set of all integers Z defined byR= [ (a,b): 2 divides a-b }(a) Show that R is an equivalence relation?(b) Find [0] and [1](c) Show that [0] and [1] form a partition of Z​

Answer 1

Step-by-step explanation:

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.

Answer 2

Answer:

vhjhLet R be a relation defined on the set of all integers Z defined byR= [ (a,b): 2 divides a-b }(a) Show that R is an equivalence relation?(b) Find [0] and [1](c) Show that [0] and [1] form a partition of Z

Step-by-step explanation:

Let R be a relation defined on the set of all integers Z defined byR= [ (a,b): 2 divides a-b }(a) Show that R is an equivalence relation?(b) Find [0] and [1](c) Show that [0] and [1] form a partition of Z