Question

1. Let X = {1,2,3,..., 100) and R {(x,y):x-y is divisible by 7). Determine whether R is an equivalence relation or not. If it is an equivalence relation, determine the partition X|R.​

Answer 1

Answer:

100 is divisible by seven

Answer 2

SOLUTION

TO CHECK

Let X = {1,2,3,..., 100}

R = {(x,y) : x - y is divisible by 7 }

Determine whether R is an equivalence relation or not

EVALUATION

Here it is given that

X = {1,2,3,..., 100}

R = {(x,y) : x - y is divisible by 7 }

CHECKING FOR REFLEXIVE

Let a ∈ X

Since a - a is divisible by 7

So (a, a) ∈ R

So R is Reflexive

CHECKING FOR SYMMETRIC

Let a, b ∈ X and (a, b) ∈ R

⇒ a - b is divisible by 7

⇒ - ( b - a ) is divisible by 7

⇒( b - a ) is divisible by 7

⇒(b, a) ∈ R

Thus (a, b) ∈ R implies (b, a) ∈ R

So R is symmetric

CHECKING FOR TRANSITIVE

Let a, b, c ∈ X

Also let (a, b) ∈ R and (b, c) ∈ R

⇒ a - b is divisible by 7 and b - c is divisible by 7

⇒( a - b + b - c ) is divisible by 7

⇒( a - c ) is divisible by 7

⇒(a, c) ∈ R

Thus (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R

R is transitive

Hence R is an equivalence relation

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