Question

Prove that the relation r in the set ={5,6,7,8,9} given by r={(a,b):|a-b|is divisible by 2} is an equivalence relation .Find all the elements related to the element b

Answer 1

Answer:

Given: Set A = { 5 , 6 , 7 , 8 , 9 }

           Relation R = { ( a , b ) : | a - b | is divisible by 2 }

To Show: R is equivalence relation.

To find: All elements related to element b

Roaster form of Relation R = { (5,5) , (6,6) , (7,7) , (8,8) , (9,9) , (5,7) , (5,9) , (6,8) , (7,9) , (7,5) , (9,5) , (8,6) , (9,7) }

An equivalence relation on a set S, is a relation on S which is  reflexive, symmetric and transitive.

1. Reflexive: For all x ∈ A, (x,x) ∈ R.

2. Symmetric: If (x,y) ∈ R then (y,x) ∈ R.

3. Transitive: If (x,y) ∈ R and (y,z) ∈ R then (x,z) ∈ R.

Thus,

1).

for  ∀ x ∈ A

⇒ |x-x|=0 is divisible by 2.

⇒ (x,z) ∈ R

⇒ R is reflexive

2).

Since in R for every (x,y) ∈ R

⇒ |x-y| is divisible by 2

⇒ |-(y-x)| is divisible by 2

⇒ |y-x| is also divisble by 2

⇒ (y,x) ∈ R

⇒ R is symmetric

3).

Since (x,y) ∈ R & (y,z) ∈ R

⇒ |x-y| is divisible by 2 &  |y-z| is divisible by 2

⇒ |x-y-(y-z)| is divisible by 2

⇒ |x-z| is divisible by 2.

⇒ R is transitive

Hence proved.

ALL elements of R are related to element b.