Your text discusses equivalence relations and their properties. Here’s a clearer breakdown:

Definition of an Equivalence Relation

A binary relation $R \subseteq A \times A$ on a set $A$ is called an equivalence relation if it satisfies the following properties:

1. Reflexivity: For every $a \in A$, $a R a$ holds.
2. Symmetry: For all $a, b \in A$, if $a R b$, then $b R a$.
3. Transitivity: For all $a, b, c \in A$, if $a R b$ and $b R c$, then $a R c$.

Equivalence Classes

For $R \subseteq A \times A$ defined as an equivalence relation, the equivalence class of an element $a$ with respect to $R$ is defined as:

$[a]_R = \{ b \in A \mid a R b \}$

Examples

• The equivalence class $[1] = \{1\}$.
• For a relation defined by $a \equiv b \mod 3$, the equivalence classes in $\mathbb{Z}$ would be:
  • $[0] = \{ \ldots, -6, -3, 0, 3, 6, \ldots \} = 3\mathbb{Z}$
  • $[1] = \{ \ldots, -5, -2, 1, 4, 7, \ldots \} = 3\mathbb{Z} + 1$
  • $[2] = \{ \ldots, -4, -1, 2, 5, 8, \ldots \} = 3\mathbb{Z} + 2$

Quotient Set

The quotient of $A$ with respect to $R$, denoted $A/R$, is the set of all equivalence classes:

$A/R = \{ [a]_R \mid a \in A \}$

Facts

• The equivalence class $[a]_R$ contains the element $a$: $a \in [a]_R$.
• If $[a]_R = [b]_R$, then $a R b$.
• If $[a]_R \cap [b]_R = \emptyset$ for $(a, b) \notin R$, then the classes are disjoint.

Partitioning

The set $A/R$ forms a partition of $A$, meaning it divides $A$ into disjoint subsets where each element of $A$ belongs to exactly one equivalence class.

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If you have questions or need more details on any of these points, feel free to ask! Here are some related questions to explore further:

1. What are some common examples of equivalence relations in mathematics?
2. How do equivalence relations relate to functions and mappings?
3. Can you explain the concept of partitions in more detail?
4. What are the implications of an equivalence relation in set theory?
5. How would you prove that a given relation is an equivalence relation?

Tip: When studying equivalence relations, always check for reflexivity, symmetry, and transitivity separately to confirm the relation's properties.