Before moving into the proof, let's understand some key terms:
In modular arithmetic, a ≡m b means that a and b leave the same remainder when divided by m. In other words, a - b is a multiple of m.
Transitivity is a property of binary relations where whenever an element a is related to an element b, and b is related to an element c, then a is also related to c.
Let's start the proof:
1. We are given in the problem that a ≡m b and b ≡m c.
2. As per the definition of modular congruence, when a and b leave the same remainder when divided by m, there exists an integer x such that: a - b = xm. This equation comes from the fact that a and b are related i.e., a - b is a multiple of m.
3. Similarly, we also know that b and c leave the same remainder when divided by m. This implies there exists a second integer y where b - c = ym. The same logic applies as before - b and c are related, so the difference between them is an integer multiple of m.
4. Now, we add these two equations together, giving us a - c = xm + ym, which simplifies to a - c = (x+y)m.
5. This says that when a and c are divided by m, they leave the same remainder. Or in other words, a ≡m c.
This develops the transitivity because it shows that if a is related to b (as given) and b is related to c (again given), then a is also related to c. Thus, the congruence modulo m is a transitive relation on the set of integers. This proves our proposition.