Q. 12

# An integer m is said to be related to another integer n if m is a multiple of n. Check if the relation is symmetric, reflexive and transitive.

Answer :

According to the question,

m is related to n if m is a multiple of n.

∀ m, n ∈ I (I being set of integers)

The relation comes out to be:

R = {(m, n): m = kn, k ∈ ℤ}

Recall that for any binary relation R on set A. We have,

R is reflexive if for all x ∈ A, xRx.

R is symmetric if for all x, y ∈ A, if xRy, then yRx.

R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.

**Check for Reflexivity:**

∀ m ∈ I

If (m, m) ∈ R

⇒ m = k m, holds.

As an integer is always a multiple of itself, So, ∀ m ∈ I, then (m, m) ∈ R.

⇒ R is reflexive.

**∴** **R is reflexive.**

**Check for Symmetry:**

∀ m, n ∈ I

If (m, n) ∈ R

⇒ m = k n, holds.

Now, replace m by n and n by m, we get

n = k m, which may or not be true.

Let us check:

If 12 is a multiple of 3, but 3 is not a multiple of 12.

⇒ n = km does not hold.

So, if (m, n) ∈ R, then (n, m) ∉ R.

∀ m, n ∈ I

⇒ R is not symmetric.

**∴** **R is not symmetric.**

**Check for Transitivity:**

∀ m, n, o ∈ I

If (m, n) ∈ R and (n, o) ∈ R

⇒ m = kn and n = ko

Where k ∈ ℤ

Substitute n = ko in m = kn, we get

m = k(ko)

⇒ m = k^{2}o

If k ∈ ℤ, then k^{2}∈ ℤ.

Let k^{2} = r

⇒ m = ro, holds true.

⇒ (m, o) ∈ R

So, if (m, n) ∈ R and (n, o) ∈ R, then (m, o) ∈ R.

∀ m, n ∈ I

⇒ R is transitive.

**∴** **R is transitive.**

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Fill in the blanks in each of the

Let the relation R be defined on the set

A = {1, 2, 3, 4, 5} by R = {(a, b) : |a^{2} – b^{2}| < 8}. Then R is given by _______.

State True or False for the statements

Every relation which is symmetric and transitive is also reflexive.

Mathematics - ExemplarState True or False for the statements

Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.

Mathematics - ExemplarState True or False for the statements

An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.

Mathematics - Exemplar