Question

If R = {m, n); m/n is a power of 5} on z-{0} then
test the relation reflexive, symmetric
and transitive.​

Answer 1

SOLUTION

TO CHECK

Check R = {m, n); m/n is a power of 5} on Z - {0} then

test the relation reflexive, symmetric and transitive.

EVALUATION

Here the given relation is

R = { (m, n) : m/n is a power of 5 } on Z - {0}

CHECKING FOR REFLEXIVE

Let m ∈ Z - {0}

Then m/m = 1 = ${5}^{0}$

So (m,m) ∈ R

So R is Reflexive

CHECKING FOR SYMMETRIC

Let m, n ∈ Z - {0} and (m, n) ∈ R

⇒m/n = ${5}^{p}$

⇒ n/m $= {5}^{ - p}$

⇒(n, m) ∈ R

Thus (m, n) ∈ R implies (n, m) ∈ R

So R is symmetric

CHECKING FOR TRANSITIVE

Let m, n, p ∈ Z

Also let (m, n) ∈ R and (n, p) ∈ R

⇒ m/n = ${5}^{r}$

and n/p = ${5}^{s}$

On multiplying we get

⇒m/p $= {5}^{r + s}$

⇒(m, p) ∈ R

∴ (m, n) ∈ R and (n, p) ∈ R implies (m, p) ∈ R

R is transitive

Hence R is an equivalence relation

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Answer 2

Step-by-step explanation:

SOLUTION

TO CHECK

Check R = {m, n); m/n is a power of 5} on Z - {0} then

test the relation reflexive, symmetric and transitive.

EVALUATION

Here the given relation is

R = { (m, n) : m/n is a power of 5 } on Z - {0}

CHECKING FOR REFLEXIVE

Let m ∈ Z - {0}

Then m/m = 1 = {5}^{0}5

0

So (m,m) ∈ R

So R is Reflexive

CHECKING FOR SYMMETRIC

Let m, n ∈ Z - {0} and (m, n) ∈ R

⇒m/n = {5}^{p}5

p

⇒ n/m = {5}^{ - p}=5

−p

⇒(n, m) ∈ R

Thus (m, n) ∈ R implies (n, m) ∈ R

So R is symmetric

CHECKING FOR TRANSITIVE

Let m, n, p ∈ Z

Also let (m, n) ∈ R and (n, p) ∈ R

⇒ m/n = {5}^{r}5

r

and n/p = {5}^{s}5

s

On multiplying we get

⇒m/p = {5}^{r + s}=5

r+s

⇒(m, p) ∈ R

∴ (m, n) ∈ R and (n, p) ∈ R implies (m, p) ∈ R

R is transitive

Hence R is an equivalence relation