If ~ be an equivalence relation on a non-empty set A, then A can be expressed as a union of mutually disjoint equivalence classes in A. Prove it?

To prove that a non-empty set A can be expressed as a union of mutually disjoint equivalence classes when ~ is an equivalence relation on A, we need to break down the concepts involved and show how they interrelate. An equivalence relation is defined by three properties: reflexivity, symmetry, and transitivity. Let's delve into the proof step by step.

Understanding Equivalence Relations

An equivalence relation ~ on a set A partitions A into distinct subsets known as equivalence classes. Each equivalence class groups elements that are considered equivalent under the relation ~. For any element a in A, the equivalence class of a, denoted [a], is defined as:

• [a] = { x ∈ A | x ~ a }

This means that [a] includes all elements x in A that are related to a by the relation ~.

Establishing the Partition

To show that A can be expressed as a union of these equivalence classes, we need to demonstrate two key points:

• Every element of A belongs to at least one equivalence class.
• Any two equivalence classes are either disjoint or identical.

Step 1: Every Element Belongs to an Equivalence Class

By the reflexivity property of equivalence relations, for any element a in A, it holds that a ~ a. Thus, the equivalence class [a] is non-empty because it contains at least the element a itself. This means that every element in A is included in at least one equivalence class.

Step 2: Equivalence Classes are Mutually Disjoint

Next, we need to show that if we take any two equivalence classes [a] and [b], they are either disjoint or the same. Assume that there exists an element x that belongs to both [a] and [b]. By the definition of equivalence classes, we have:

• x ~ a (since x ∈ [a])
• x ~ b (since x ∈ [b])

Using the symmetry and transitivity properties of the equivalence relation, we can conclude:

• From x ~ a and x ~ b, we can say a ~ b (by transitivity).

This implies that every element in [a] is also in [b], leading us to conclude that [a] = [b]. Therefore, the equivalence classes are either disjoint or identical.

Combining the Results

Since every element of A belongs to at least one equivalence class and any two equivalence classes are either disjoint or the same, we can conclude that the set A can indeed be expressed as a union of these equivalence classes:

• A = ∪ [a] for all a ∈ A

In summary, we have shown that A can be partitioned into equivalence classes that are mutually disjoint, confirming the statement. This property is fundamental in various areas of mathematics, including set theory and algebra, as it allows us to categorize elements based on their relationships.