Question

The relation “congruence modulo m” is
A. reflexive only
B.transitive only
C. symmetric only
D. an equivalence relation

Answer 1

Hint: We know that if two numbers have the property that their difference is integrally divisible by a number then they are said to be congruent modulo. Use this principle to get the answer.

Complete step-by-step answer:
Now first of all let us assume the relation of congruence modulo as R.
And we know that for a congruence modulo the difference must be divisible by the number.
So, xRy = x – y is divisible by m.
And now xRx because x – x is also divisible by m.
So, from the above we can say that the relation R is a reflexive relation.
And if x – y is divisible by m then y – x is also divisible by m. (as here it isn’t mentioned that m will be a positive or a negative integer )
So, now we can say that R is also a symmetric relation.
And now xRy an yRz is also equals to
\[x - y = {a_1}m\]and \[y - z = {a_{_2}}m\]
Now adding the above two equations
(x – y) + ( y – z ) = \[{a_1}m + {a_{_2}}m\]
So, x – z = \[({a_1} + {a_{_2}})m\]
Now from above we can say that R is a transitive relation also.
So, R is a reflexive , symmetric and transitive relation and when a relation belongs to all the three relations then it is called an equivalence relation.
Hence D is a correct option.

Note :- . A reflexive relation belongs to itself only. A symmetric relation is a type of binary relation and transitive relation is a homogeneous relation whereas equivalence relation is a relation which includes all the three relations.