Question

A relation R defined in the set N of natural numbers as R = {(a,b) : |a-b| is even } , is an equivalence relation. write the set of numbers which are related with no 1 according to given relation. ​

Answer 1

Answer:

Given A={1,2,3,4,5} and R={(a,b):∣a−b∣is even}

To prove that it is equivalent relation we need to prove that R is reflexive, symmetric and transitive.

(i) Reflexive:

Let aϵA

then ∣a−a∣=0 is an even number

∴(a,a)ϵR,∀aϵA

∴R is reflexive

(ii) Symmetric

Let a,bϵA

∀(a,b)ϵR⇒∣a−b∣ is even

⇒∣−(b−a)∣ is even

⇒∣b−a∣ is even

⇒∣b−a∣ϵR

or (b,a)ϵR

∴R is symmetric

(iii) Transitive

Let a,b,cϵA

∀(a,b)ϵR and (b,c)ϵR

we have ∣a−b∣ is even and ∣b−c∣ is even

⇒a−b is even and b−c is even

⇒a−b is even and b−c is even

⇒(a−b)+(b−c) is even

⇒a−c is even

⇒∣a−c∣ is even ⇒(a,c)ϵR∴R is transitive

∴R is an equivalence relation.

Step-by-step explanation: