Question

Suppose \(P\) is a partition of a set \(A .\) Define a relation \(R\) on \(A\) by declaring \(x R y\) if and only if \(x, y \in X\) for some \(X \in P\). Prove \(R\) is an equivalence relation on \(A\). Then prove that \(P\) is the set of equivalence classes of \(R\).

Short answer

The relation \(R\) is an equivalence relation since it meets all the necessary properties (reflexivity, symmetry, and transitivity). Furthermore, the partition \(P\) on the set \(A\) corresponds to the set of equivalence classes of \(R\).

Step-by-step solution

Prove \(R\) is an equivalence relation

To prove the relation \(R\) is an equivalence relation, we need to show it is reflexive, symmetric, and transitive.\n 1.1 Reflexivity: For any element \(a\) in \(A\), from the definition of the partition, \(a\) belongs to the same subset \(X\) under \(P\). Therefore, \(a R a\) holds because \(a,a \in X\).\n 1.2 Symmetry: If a relation \(a R b\) holds, then both \(a\) and \(b\) belong to some subset \(X\) (in \(P\)). By definition, \(b\) and \(a\) also belong to the same subset, and therefore \(b R a\) holds.\n 1.3 Transitivity: If a relation \(a R b\) and \(b R c\) hold, then \(a\), \(b\), and \(c\) belong to the same subset \(X\). Hence, \(a R c\) holds.

Prove \(P\) is equal to the set of equivalence classes of \(R\)

Given a partition \(P\) on \(A\), we have already established an equivalence relation \(R\), and now we want to show that \(P\) corresponds to the set of equivalence classes of \(R\). Since every element \(a\) belongs to its own equivalence class \([a]\), the collection of all equivalence classes covers \(A\). As \(R\) is an equivalence relation, these equivalence classes must partition \(A\). Hence, the collection must correspond to \(P\). This verifies that the set \(P\) equals the set of the equivalence classes of \(R\).

Key concepts

Partition of a Set

When we talk about the partition of a set, imagine dividing a whole into exclusive and exhaustive pieces. Each piece is called a block or part of the partition. This partitioning means you can take any element from the set, and it will fit into one and only one of these pieces.

The key properties are:

• No part is empty.
• Every element of the whole set is included in a part.
• Parts are mutually exclusive, meaning they don't overlap; no element can be found in more than one part.

Think of partitioning as cutting a cake into slices - every slice is a part, no slice is left empty, every crumb of the cake is in a slice, and no crumb exists in more than one slice.

Reflexive Relation

A reflexive relation in mathematics is like giving yourself a high five - it must always work! In a set, if every element is related to itself, we say the relation on this set is reflexive. More formally, a relation R on a set A is reflexive if, for every element a in A, it's always true that a R a.

This concept is fundamental for equivalence relations because it insists on the inclusion of certain paired elements – primarily, each element with itself – within the relationship.

Symmetric Relation

The idea of a symmetric relation is like a two-way street; if you can go from point A to point B, you must be able to return from B to A. For a relation R on a set A, this symmetry happens if whenever an element a is related to an element b (writing it as a R b), it's also true that b is related to a (b R a).

If a friend is considered your friend (symmetric, isn't it?), then we have a perfect example of a symmetric relation. In the context of an equivalence relation, symmetry ensures that the relation doesn't only go one way.

Transitive Relation

A transitive relation adds a layer to our understanding of relations. It deals with connections over a chain of elements. If for some elements a, b, and c in a set, the fact that a is related to b (a R b), and b is related to c (b R c), implies that a must be related to c (a R c), then the relation R is transitive.

Think of it as a friend-of-a-friend scenario - if Alice is friends with Bob, and Bob with Charlie, then Alice should be considered friends with Charlie. In relations, transitivity helps us extend linkages within the set.

Equivalence Classes

Now let's get into the VIP section of relations: equivalence classes. Each member of a set with an equivalence relation can lead a group - its equivalence class. This class consists of all elements that are related to it. For an element a in A, its equivalence class, denoted by [a], is the set of all elements in A that are related to a.

The beautiful thing about equivalence classes within a set is that they form a partition of that set. Every element belongs to one and only one equivalence class, so they satisfy the properties needed for a set to be partitioned. There's no overlap, and every element gets included. It's an exclusive club where relation R defines the membership.