Question

Prove that the relation R on the set Z-{0}, defined as R = {(a,b):a÷b is a power of 5} is
an equivalence relation.​

Answer 1

SOLUTION

TO PROVE

The relation R on the set Z- {0}, defined as

R = {(a,b):a÷b is a power of 5}

is an equivalence relation

EVALUATION

Here the given relation is R on the set Z- {0}, defined as

R = { (a,b) : a÷b is a power of 5 }

CHECKING FOR REFLEXIVE

Let a ∈ Z- {0}

$\sf{a \div a = 1 = {5}^{0} }$

Hence a ÷ a is a power of 5

Hence (a, a) ∈ R for all a ∈ Z- {0}

Hence R is Reflexive

CHECKING FOR SYMMETRIC

Let a, b ∈ Z- {0} such that (a, b) ∈ R

(a, b) ∈ R

⟹ a ÷ b is a power of 5

$\sf{ \implies \: a \div b = {5}^{n} \: \: \: for \: some \: n }$

$\sf{ \implies \: b \div a = {5}^{ - n} \: \: \: for \: some \: n }$

⟹ b ÷ a is a power of 5

⟹ (b, a) ∈ R

Hence R is symmetric

CHECKING FOR TRANSITIVE

Let a, b, c ∈ Z- {0} such that (a, b) ∈ R & (b, c) ∈ R

Now (a, b) ∈ R & (b, c) ∈ R

⟹ a ÷ b is a power of 5 & b ÷ c is a power of 5

⟹ a ÷ c is a power of 5

⟹ (a, c) ∈ R

Hence R is transitive

Hence R is an equivalence Relation

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