1. The problem asks us to explain what an equivalence relation is and its use with an example. 2. An equivalence relation on a set is a relation that satisfies three properties: - Reflexive: Every element is related to itself, i.e., for every $a$, $a \sim a$. - Symmetric: If $a$ is related to $b$, then $b$ is related to $a$, i.e., if $a \sim b$, then $b \sim a$. - Transitive: If $a$ is related to $b$ and $b$ is related to $c$, then $a$ is related to $c$, i.e., if $a \sim b$ and $b \sim c$, then $a \sim c$. 3. Equivalence relations are useful because they partition a set into equivalence classes, where all elements in the same class are related. 4. Example: Consider the set of integers $\mathbb{Z}$ and define the relation $a \sim b$ if and only if $a - b$ is divisible by 3. - Reflexive: $a - a = 0$ is divisible by 3, so $a \sim a$. - Symmetric: If $a \sim b$, then $a - b$ is divisible by 3, so $b - a = -(a - b)$ is also divisible by 3, so $b \sim a$. - Transitive: If $a \sim b$ and $b \sim c$, then both $a - b$ and $b - c$ are divisible by 3, so their sum $a - c = (a - b) + (b - c)$ is divisible by 3, hence $a \sim c$. 5. Thus, this relation is an equivalence relation and it partitions the integers into classes like $\{...,-6,-3,0,3,6,...\}$, $\{...,-5,-2,1,4,7,...\}$, and $\{...,-4,-1,2,5,8,...\}$ depending on remainder modulo 3. 6. This helps in classifying integers into groups with similar properties modulo 3, which is valuable in number theory and algebra.