The statement you've provided illustrates a fundamental connection between equivalence relations and set partitions. Let's break this down:

1. Equivalence Relations and Partitions:

• An equivalence relation on a set $A$ is a binary relation that satisfies three properties:
  1. Reflexivity: $a \sim a$ for all $a \in A$.
  2. Symmetry: If $a \sim b$, then $b \sim a$.
  3. Transitivity: If $a \sim b$ and $b \sim c$, then $a \sim c$.
• When an equivalence relation $\sim$ is defined on $A$, the set $A$ can be divided into equivalence classes. Each equivalence class is a subset of $A$ where all elements are equivalent to each other under $\sim$.

2. Decomposition into Disjoint Subsets:

• The equivalence classes formed by an equivalence relation are mutually disjoint (no element belongs to more than one equivalence class) and their union is the entire set $A$.
• In other words, the set $A$ is decomposed into a union of mutually disjoint, nonempty subsets, where each subset corresponds to an equivalence class.

3. Conversely, Constructing an Equivalence Relation:

• If we start with a decomposition of $A$ into mutually disjoint, nonempty subsets $\{A_1, A_2, \dots, A_n\}$, we can define an equivalence relation $\sim$ on $A$ as follows:
  • For $a, b \in A$, define $a \sim b$ if and only if $a$ and $b$ belong to the same subset $A_i$ for some $i$.
• This relation is an equivalence relation because:
  1. Reflexivity: Since each element belongs to some subset $A_i$, it is related to itself.
  2. Symmetry: If $a$ and $b$ are in the same subset, $a \sim b$ and $b \sim a$.
  3. Transitivity: If $a \sim b$ and $b \sim c$, all three elements $a$, $b$, and $c$ must belong to the same subset $A_i$, hence $a \sim c$.

Conclusion:

The connection between equivalence relations and partitions of a set is a deep and powerful concept in mathematics. The distinct equivalence classes of an equivalence relation provide a partition of the set, and conversely, any partition of a set determines an equivalence relation.

Would you like further details on this topic or have any questions?

Related Questions:

1. What is the definition of an equivalence class, and how can it be identified?
2. How does the concept of set partitions relate to other mathematical areas like topology or group theory?
3. Can you provide an example of an equivalence relation and its corresponding partition on a specific set?
4. What are the applications of equivalence relations in real-world problems?
5. How do equivalence relations relate to the concept of symmetry in mathematics?
6. Can two different equivalence relations on the same set result in the same partition?
7. What is the significance of nonempty subsets in the context of partitions?
8. How does the notion of disjoint subsets ensure the properties of an equivalence relation?

Tip:

Remember, understanding the relationship between equivalence relations and partitions can simplify complex problems by reducing them to manageable, smaller subsets.