Question

The relation of "congruence modulo" is :

• A. reflexive only
• B. symmetric only
• C. transitive only
• D. an equivalence relation

Answer 1

Answer: (D)

Congruence modulo: An integer ‘a’ is said to be congruent to another integer ‘b’ module ‘m’ if a - b is divisible by m and expressed as: a $\equiv$ b (mod m)
The integer m is called the modulus of the congruence.
Now suppose a, b, c $\in$ I (congruence of modulo)
Let a $\equiv$ b (mod m)
Since a $\equiv$ a (mod m) and o $\doteq$ m
so the function is Reflexive.
Now if a $\equiv$ b (mod m) then
Also b $\equiv$ a (mod m) because (a - b) divides by m, then b - a also divides by m. Thus function is symmetric.
Now If a $\equiv$ b (mod m) and b $\equiv$ c (mod m)
$\Rightarrow \frac{a-b}{m}$ and $\frac{b-c}{m};$ So $\frac{a-c}{m}$
$\therefore a\equiv c$ (mod m) thus function is Transitive.
Thus the relation is Equivalence Relation.