To prove that the relation $a \equiv b \pmod{m}$ is an equivalence relation, we must show that the relation satisfies three properties of an equivalence relation:

1. Reflexivity: For any integer $a$, we need to show that $a \equiv a \pmod{m}$.

2. Symmetry: For any integers $a$ and $b$, if $a \equiv b \pmod{m}$, we need to show that $b \equiv a \pmod{m}$.

3. Transitivity: For any integers $a$, $b$, and $c$, if $a \equiv b \pmod{m}$ and $b \equiv c \pmod{m}$, we need to show that $a \equiv c \pmod{m}$.

1. Reflexivity

We need to show that $a \equiv a \pmod{m}$. By definition of congruence modulo $m$, $a \equiv b \pmod{m}$ means that $m$ divides the difference $a - b$, i.e., $a - b = km$ for some integer $k$.

For $a \equiv a \pmod{m}$, the difference $a - a = 0$, which is clearly divisible by $m$. Therefore, $a \equiv a \pmod{m}$, satisfying reflexivity.

2. Symmetry

We need to show that if $a \equiv b \pmod{m}$, then $b \equiv a \pmod{m}$.

By the definition of congruence, if $a \equiv b \pmod{m}$, then $m \mid (a - b)$. This means there exists an integer $k$ such that $a - b = km$.

Now, consider $b - a$. We have: $b - a = -(a - b) = -km$ Since $-km$ is also divisible by $m$, it follows that $m \mid (b - a)$. Thus, $b \equiv a \pmod{m}$, proving symmetry.

3. Transitivity

We need to show that if $a \equiv b \pmod{m}$ and $b \equiv c \pmod{m}$, then $a \equiv c \pmod{m}$.

From the assumption $a \equiv b \pmod{m}$, we know that $m \mid (a - b)$, so there exists an integer $k_1$ such that: $a - b = k_1 m$

Similarly, from $b \equiv c \pmod{m}$, we know that $m \mid (b - c)$, so there exists an integer $k_2$ such that: $b - c = k_2 m$

Now, consider the difference $a - c$: $a - c = (a - b) + (b - c) = k_1 m + k_2 m = (k_1 + k_2) m$ Since $a - c$ is a multiple of $m$, we conclude that $m \mid (a - c)$, so $a \equiv c \pmod{m}$. This proves transitivity.

Conclusion

Since we have shown that the relation $a \equiv b \pmod{m}$ satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.

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Do you need further clarification or examples for any part of this proof?

Here are some related questions you might find useful:

1. How do you define congruence modulo $m$ for integers?
2. What are some examples of equivalence relations in other contexts (e.g., set theory)?
3. Can you prove that congruence modulo $m$ is an equivalence relation in any other number system, such as real numbers?
4. What is the relationship between the equivalence classes of $a$ and $b$ under modulo $m$?
5. How does the proof of transitivity differ if we are working with other types of relations, such as partial orders?

Tip: When working with equivalence relations, it's helpful to always verify the three properties (reflexivity, symmetry, and transitivity) in sequence for clarity and correctness.