Equivalence Class { Philosophy Index }

Philosophy Index

Philosophy Index

Philosophy Index is a site devoted to the study of philosophy and the philosophers who conduct it. The site contains a number of philosophy texts, brief biographies and introductions to philosophers and explanations on a number of topics. Accredited homeschooling online at Northgate Academy.

Philosophy Index is a work in progress, a growing repository of knowledge. It outlines current philosophical problems and issues, as well as an overview of the history of philosophy. The goal of this site is to present a tool for those learning philosophy either casually or formally, making the concepts of philosophy accessible to anyone interested in researching them. WOLI offers immigration law course online - fully accredited. ACE credits online at EES.

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Subset

In set theory, an equivalence class is a set of all objects that are related to a specific object by some equivalence relation.

For example, if R is an equivalence relation on a set S, and aS, then the equivalence class [a] is the set of all elements of S that are R-related to a.

[a] = { x|Rxa }

The notation for equivalence classes varies among logic texts. On this site, the notation [a] is used to denote an equivalence class of the element a. If there are multiple relations being considered, a subscript of the specific relation may be added: [a]R. Some texts will place a line over the element name instead, using the notation a to indicate an equivalence class.

Examples

Suppose that S is a set and R is an equivalence class on that set, such that:

Then the following are true:

  1. [a] = { a }
  2. [b] = { b, c }
  3. [c] = { b, c }
  4. [e] = { b, c, e }

(1) is true because the only element R-related to a (that is, the only x such that <x, a> ∈ R) is a itself. (2) and (3) are true because b and c are both R-related to themselves, and to each other, but nothing else is related to them. Similarly, in (4), b and c are both R-related to e, and e is R-related to itself.

With an equivalence relation on a set S, for every xS, x ∈ [x]. This is because equivalence relations are reflexive — that is, every element is related to itself, and is therefore a member of its own equivalence class.