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Equivalence Class: Definition

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An equivalence class is the name that we give to the subset of S which includes all elements that are equivalent to each other. “Equivalent” is dependent on a specified relationship, called an equivalence relation. If there’s an equivalence relation between any two elements, they’re called equivalent.

A simple equivalence class might be defined with an equals sign. We could say

‘The equivalence class of a consists of the set of all x, such that x = a’.

In other words, any items in the set that are equal belong to the defined equivalence class. This set seems like a rather trivial set, but there are other equivalence relations which make things rather more interesting. We’ll look at a few of the simpler ones below.

Properties of an Equivalence Class

Let’s represent our equivalence relation by ~ (it may really be = or any number of other things). The relation ~ is an equivalence relation if and only if:

  1. It is reflexive: any a in X must always be equivalent to itself; we can write this as a ~ a
  2. It is symmetric: Suppose a, b are in X. Then, if a is equivalent to b, b will also be equivalent to a. We can write this as if a ~ b, b ~ a
  3. It is transitive: Let a, b, and c be elements of X. Then, if a is equivalent to b, and b is equivalent to c, a will also be equivalent to c. We can write this as: for a, b, c in X; if a ~ b and b ~ c it follows that a ~ c

Once we’ve checked to make sure our relation ~ satisfies the three properties above, we can write the definition of an equivalence class of an element a like this

[a] = { x Ε X | a ~ x}

Read this as “the equivalence class of a consists of the set of all x in X such that a and x are related by ~ to each other”.

Examples of Equivalence Classes

Suppose X was the set of all children playing in a playground. Then if ~ was an equivalence relation for ‘of the same age’, one equivalence class would be the set of all 2-year-olds, and another the set of all 5-year-olds.

If X was the set of all polygons, and the equivalence relation ~ was defined as ‘ with the same number of sides’, examples of equivalences classes would be the set of all triangles, the set of all quadrilaterals, the set of all pentagons, etc.,

If X is the set of all integers, we can define the equivalence relation ~ by saying ‘a ~ b if and only if ( a – b ) is divisible by 9’. Then the equivalence class of 4 would include -32, -23, -14, -5, 4, 13, 22, and 31 (and a whole lot more).

References

Arnold, Jimmy. An Introduction to Mathematical Proofs. Chapter 5: Equivalence Relations and Equivalence Classes
retrieved from http://www.math.vt.edu/people/elder/Math3034/book/3034Chap5.pdf on March 18, 2018

Watkins, Thayer. Equivalence.
retrieved from http://www.sjsu.edu/faculty/watkins/equivalence.htm on March 18, 2018

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