The problem statement in the image is as follows:

Consider the set $\mathbb{Z}$ of integers and an integer $m > 1$. We say that $a$ is congruent to $b$ modulo $m$, written $a \equiv b \ (\text{mod} \ m)$, if $m$ divides $a - b$. Show that this defines an equivalence relation on $\mathbb{Z}$.

Solution:

To prove that the relation $a \equiv b \ (\text{mod} \ m)$ is an equivalence relation, we must show that it satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

1. Reflexivity:

   • For any integer $a \in \mathbb{Z}$, we need to show that $a \equiv a \ (\text{mod} \ m)$.
   • $a \equiv a \ (\text{mod} \ m)$ means that $m$ divides $a - a$.
   • Since $a - a = 0$, and $m$ divides 0, this property holds true for any $a$.
   • Therefore, the relation is reflexive.

2. Symmetry:

   • We need to show that if $a \equiv b \ (\text{mod} \ m)$, then $b \equiv a \ (\text{mod} \ m)$.
   • If $a \equiv b \ (\text{mod} \ m)$, it means that $m$ divides $a - b$, i.e., $a - b = km$ for some integer $k$.
   • This can be rewritten as $b - a = -km$, and since $-k$ is also an integer, $m$ divides $b - a$.
   • Therefore, $b \equiv a \ (\text{mod} \ m)$, so the relation is symmetric.

3. Transitivity:

   • We need to show that if $a \equiv b \ (\text{mod} \ m)$ and $b \equiv c \ (\text{mod} \ m)$, then $a \equiv c \ (\text{mod} \ m)$.
   • If $a \equiv b \ (\text{mod} \ m)$, then $m$ divides $a - b$, i.e., $a - b = km$ for some integer $k$.
   • If $b \equiv c \ (\text{mod} \ m)$, then $m$ also divides $b - c$, i.e., $b - c = lm$ for some integer $l$.
   • Adding these two equations gives $(a - b) + (b - c) = km + lm = (k + l)m$, which implies $a - c = (k + l)m$.
   • Since $k + l$ is an integer, $m$ divides $a - c$, and thus $a \equiv c \ (\text{mod} \ m)$.
   • Therefore, the relation is transitive.

Since the relation $a \equiv b \ (\text{mod} \ m)$ satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation on the set $\mathbb{Z}$.

Summary:

The relation $a \equiv b \ (\text{mod} \ m)$ is an equivalence relation on the set of integers $\mathbb{Z}$ because it satisfies the three necessary properties: reflexivity, symmetry, and transitivity.

Would you like more details or have any questions?

Related Questions:

1. What are some real-life applications of equivalence relations?
2. How does modular arithmetic play a role in computer science?
3. Can you provide an example of an equivalence class under this relation?
4. How does the concept of congruence modulo $m$ relate to cyclic groups?
5. How is the Chinese Remainder Theorem related to modular arithmetic?

Tip:

Remember that an equivalence relation partitions a set into disjoint equivalence classes. Understanding this concept is fundamental in various fields such as algebra and geometry.