³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980399067³2ip³2129.97.13.47³2data³2newauthor³31³3minor³30³3text³3An equivalence relation ~ on a set X is a RelatioN satisfying the following conditions: for every a,b,c in X,

    a~a                                 (reflexive property)
    If a~b, then b~a                    (symmetric property)
    If a~b, b~c, then a~c               (transitive property)

Given any x in X, we define the equivalence class of x to be the set [x] = {y in G : x~y}.  The set of all such equivalence classes is called the quotient X/~.  These form a partition of X, and conversely any partition of X is a quotient X/~ for some equivalence relation ~.

A trivial example of an equivalence relation is equality.  X = X/(=) for any set X.

³3summary³3good place to put the definition of a quotient set³2tscreate³2980399067³2keepts³2980399382³2id³21076³2host³2cnts6p47.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980399382³2ip³2129.97.13.47³2data³2text³3An equivalence relation ~ on a set X is a RelatioN satisfying the following conditions: for every a,b,c in X,

    a~a                                 (reflexive property)
    If a~b, then b~a                    (symmetric property)
    If a~b, b~c, then a~c               (transitive property)

Given any x in X, we define the equivalence class of x to be the set [x] = {y in G : x~y}.  The equivalence classes form a partition of X, and conversely every partition of x corresponds to some unique equivalence relation ~.

The set of all equivalence classes is denoted X/~, and called a quotient set.  In cases where X has some additional structure preserved under ~, the quotient naturally becomes an object of the same type; the map that sends x to [x] is then a HomoMorphism.
³3minor³30³3newauthor³30³3summary³3good place to put the definition of a quotient set³2tscreate³2980399067³2keepts³2980401477³2id³21076³2host³2cnts6p47.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980401477³2ip³2129.97.182.64³2data³2newauthor³30³3minor³31³3text³3An equivalence relation ~ on a set X is a RelatioN satisfying the following conditions: for every a,b,c in X,

    a~a                                 (reflexive property)
    If a~b, then b~a                    (symmetric property)
    If a~b, b~c, then a~c               (transitive property)

Given any x in X, we define the equivalence class of x to be the set [x] = {y in G : x~y}.  The equivalence classes form a partition of X, and conversely every partition of x corresponds to some unique equivalence relation ~.

The set of all equivalence classes is denoted X/~, and called a quotient set.  In cases where X has some additional structure preserved under ~, the quotient naturally becomes an object of the same type; the map that sends x to [x] is then a HomoMorphism.

Equality always defines an equivalence relation, and in fact it is minimal under implications.  X = X/=.

³3summary³3*³2tscreate³2980399067³2keepts³2980402084³2id³21076³2host³2cnts2p64.uwaterloo.ca³2revision³23³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980402084³2ip³2129.97.182.64³2data³2text³3An equivalence relation ~ on a set X is a MathematicalRelation satisfying the following conditions: for every a,b,c in X,

    a~a                                 (reflexive property)
    If a~b, then b~a                    (symmetric property)
    If a~b, b~c, then a~c               (transitive property)

Given any x in X, we define the equivalence class of x to be the set [x] = {y in G : x~y}.  The equivalence classes form a partition of X, and conversely every partition of x corresponds to some unique equivalence relation ~.

The set of all equivalence classes is denoted X/~, and called a quotient set.  In cases where X has some additional structure preserved under ~, the quotient naturally becomes an object of the same type; the map that sends x to [x] is then a HomoMorphism.

Equality always defines an equivalence relation, and in fact it is minimal under implications.  X = X/=.

³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980399067³2keepts³2980408127³2id³21076³2host³2cnts2p64.uwaterloo.ca³2revision³24³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980408127³2ip³2129.97.182.33³2data³2newauthor³30³3minor³31³3text³3An equivalence relation ~ on a set X is a MathematicalRelation satisfying the following conditions: for every a,b,c in X,

    a~a                                 (reflexive property)
    If a~b, then b~a                    (symmetric property)
    If a~b, b~c, then a~c               (transitive property)

Given any x in X, we define the equivalence class of x to be the set [x] = {y in G : x~y}.  The equivalence classes form a partition of X, and conversely every partition of x corresponds to some unique equivalence relation ~.

The set of all equivalence classes is denoted X/~, and called a quotient set.  In cases where X has some additional structure preserved under ~, the quotient naturally becomes an object of the same type; the map that sends x to [x] is then a HomoMorphism.

Equality always defines an equivalence relation, and in fact it is minimal.  X = X/=.

³3summary³3*³2tscreate³2980399067³2keepts³2980730181³2id³21076³2host³2cnts2p33.uwaterloo.ca³2revision³25