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 }


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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