The notion of equivalence classes is useful for constructing sets
out of already constructed ones. The set of all equivalence classes
in X given an equivalence relation ~ is usually denoted as X / ~
and called the quotient set of X by ~. This operation can be
thought of the act of "dividing" the input set by the equivalence
relation, hence both the name "quotient," and the notation, which
are both reminiscent of division. One way in which the quotient set
resembles division is that if X is finite and the equivalence
classes are all equinumerous, then the order of X/~ is the quotient
of the order of X by the order of an equivalence class. The
quotient set is to be thought of as the set X with all the
equivalent points identified.
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