Question
The relation ‘‘ congruence modulo $$m$$ ’’ is :
A.
reflexive only
B.
transitive only
C.
symmetric only
D.
an equivalence relation
Answer :
an equivalence relation
Solution :
If $$R$$ be the relation, $$x\,R\,y \Leftrightarrow x - y$$ is divisible by $$m.$$
$$x\,R\,x$$ because $$x-x$$ is divisible by $$m.$$ So, $$R$$ is reflexive.
$$x\,R\,y \Rightarrow y\,R\,x.$$ So, $$R$$ is symmetric.
$$x\,R\,y$$ and $$y\,R\,z \Rightarrow x - y = {k_1}m,\,\,y - z = {k_2}m$$
$$\therefore x - z = \left( {{k_1} + {k_2}} \right)m.$$ So, $$R$$ is transitive.
As $$R$$ is reflexive, symmetric and transitive, it is an equivalence relation.