# Difference between revisions of "Equivalence class"

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+ | <noinclude>[[Category:Relations]]</noinclude> | ||

Consider the following [[equivalence relation]] <math>R</math> on the set <math>A := \{a,b,c,d,e,f\}</math>: | Consider the following [[equivalence relation]] <math>R</math> on the set <math>A := \{a,b,c,d,e,f\}</math>: | ||

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In this example, the three groups are <math>\{a,b,c\}</math>, <math>\{d,e\}</math> and <math>\{f\}</math>. | In this example, the three groups are <math>\{a,b,c\}</math>, <math>\{d,e\}</math> and <math>\{f\}</math>. | ||

− | In fact, [[Claim:A/R partitions A|this is a general phenomenon]], and the groups often represent something important. For example, | + | In fact, [[Claim:A/R partitions A|this is a general phenomenon]], and the groups often represent something important. For example, {{:Example:Families are equivalence classes}} |

In general, the groups are referred to as [[equivalence class]]es. Formally, we have the following definition: | In general, the groups are referred to as [[equivalence class]]es. Formally, we have the following definition: | ||

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In the above example, we have three [[equivalence class]]es: <math>[[⟦a⟧]] = \{a,b,c\} = [[⟦a⟧|⟦b⟧]] = [[⟦a⟧|⟦c⟧]]</math>, <math>[[⟦a⟧|⟦d⟧]] = \{d,e\} = [[⟦a⟧|⟦e⟧]]</math>, and <math>[[⟦a⟧|⟦f⟧]] = \{f\}</math>. | In the above example, we have three [[equivalence class]]es: <math>[[⟦a⟧]] = \{a,b,c\} = [[⟦a⟧|⟦b⟧]] = [[⟦a⟧|⟦c⟧]]</math>, <math>[[⟦a⟧|⟦d⟧]] = \{d,e\} = [[⟦a⟧|⟦e⟧]]</math>, and <math>[[⟦a⟧|⟦f⟧]] = \{f\}</math>. | ||

+ | |||

+ | You can see in this example that [[Claim:⟦a⟧=⟦b⟧ if and only if aRb|⟦a⟧=⟦b⟧ if and only if aRb]]; this is always the case. | ||

{{:A mod R}} | {{:A mod R}} |

## Latest revision as of 13:42, 20 February 2020

Consider the following equivalence relation on the set :

Notice that the elements of are split up into three groups, with everything in one group related to everything else in the same group, and nothing in one group related to anything in any other group.

In this example, the three groups are

, and .In fact, this is a general phenomenon, and the groups often represent something important. For example, if , then the groups of related people would be called "families". Two people are in the same family if and only if they are related to each other. were a set of people and R was the is-related-to relation

In general, the groups are referred to as equivalence classes. Formally, we have the following definition:

**Equivalence class**

In other words,

**Note:** I usually use the symbols [ and ] instead of ⟦ and ⟧, but the wiki syntax makes this difficult. You may use either notation.

In the above example, we have three equivalence classes: , , and .

You can see in this example that ⟦a⟧=⟦b⟧ if and only if aRb; this is always the case.

**A/R**

Continuing the above example, we have in , but we don't need to list it separately, since it is equal to .

. Note that is alsoOne more piece of terminology:

**representative**