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.

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 }

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

Substitute n = ko in m = kn, we get

m = k(ko)

m = k2o

If k , then k2 .

Let k2 = 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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