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Partitions, equivalence relations and equivalence classes.

It is possible to divide sets into distinct partitions. For example, the set {Junior, Sunny, Felipe} can be divided into distinct subsets which do not share the same members. We can divide this group into {{Junior, Felipe}, {Sunny}}. Such a division is known as a partition. It would not be a partition, however, if we wrote {{Junior, Felipe}, {Sunny, Felipe}}, because Felipe is a member of both groups. In order for the division of the set into subsets to be a genuine partition, Felipe would have to get out of one of the groups. It is possible to represent partitions as members of a single set through the operation of union. For example, The union of {{Felipe, Junior}, {Sunny}} = {Felipe, Junior, Sunny}.

Having discussed the properties of reflexivity, symmetry, anti-symmetry and transitivity, we are prepared to discuss the concept of an equivalence relation. An equivalence relation is one "that is reflexive, symmetric, and transitive. Philosophers have long been very interested in equivalence relations. Two particularly interesting equivalence relations are identity and indiscernibility"(Steinhart, p. 27).

Let's briefly review the concepts of transitivity, reflexivity and symmetry, which we have covered in previous articles, and see how the co-ordination of all these properties issue in an equivalence relation:

"If F denotes an attribute of a thing, such as its color, shape, or weight, then any relation of the form is-the-same-F-as is an equivalence relation. Let's consider the relation is-the-same-color-as. Obviously, a thing is the same color as itself. So is-the-same-color-as is reflexive. For any x and y, then y is the same color as x. So is-the-same-color-as is symmetric. For any x, y, and z, if x is the same color as y, and y is the same color as z, then x is the same color as z. So is-the-same-color-as is transitive"(Steinhart, p. 27).

When we partition an equivalence relation into distinct subsets, we have what is called an equivalence class. For example, suppose we were to generate a set H, among whom the symmetric relation "has-the-same-emotion-as" obtained.

{R1, R2, L1, L2, P1, P2, J1).

L1 and L2 are all elements that are happy. R1 and R2 are all sad. Both of these would be instances of equivalence classes. The subset Y of all things that are happy would include the equivalence class L1 and L2. The subset Z, which partitions the set into the subsets of all things that are sad, would include R1, R2. Note that for the numbers affixed to the sets, these are ordinarily written as subscripts. Be sure you are able to recognize this notation.

Let's look at how we would describe an equivalence relation in formal notation:

[x]R = {y | y bears equivalence relation R to x}.

Note that the "R" stands for "relation," and will ordinarily be written as a subscript. We do not necessarily have to write the R, provided that the context makes it clear that a relation is being denoted (Steinhart, p. 28).

[x] = { y | y ∈ H & y is characterized by the same emotion as x}

This signifies that an equivalence relation obtains between elements x and y, since they are both members of H which are characterized by the same emotion. We can also signify this relation of equivalence by writing "the happy things = [L1] = [L2] = {L1, L2}."

The partition of H by is-characterized-by-the-same-emotion-as = { [x] | x ∈ H}.

Steinhart notes that equivalence classes are all mutually disjoint. This means that they do not possess any of the same elements. Therefore, A ∩ B = {}. The reason for this is that all equivalence classes, by definition of the operation, are characterized by elements which possess the properties specified.

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.

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