2
Q. No. 18 Let R be a relation on Q, defined by R={(a,b)/a,b EQ and a-be Z} .Show that R is
an equivalence relation.

November 29, 2021

By Hadley

1 thought on “2<br />Q. No. 18 Let R be a relation on Q, defined by R={(a,b)/a,b EQ and a-be Z} .Show that R is<br />an equivalence relation.<br”

1. Answer:

   R is equivalence relation.

   Step-by-step explanation:

   Given : R = {(a,b) : a, b € Q and a- b € Z}.

   1. Let a € Q . Then , a – a = 0 € Z.

   .;. (a,a) € R for all a € Q.

   So, R is reflexive.

   2. (a,b) € R => (a-b) € Z, (a -b ) is an integer

   => – (a-b) is an integer

   => (b-a) is an integer

   => (a,b) € R.

   .;. R is symmetric.

   3. (a,b) € R and (b,a) € R

   = (a- b) is an integer and (b-c) is an integer

   = {(a-b) + (b-c)} is an integer

   = (a-c) is an integer

   = (a,c) € R.

   Thus (a,b) € R and (b,c) € R = (a,c) € R.

   .;. R is transitive.

   Reply