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Left cosets partition a group

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Equivalence of statements
==Equivalence of statements==
These statements are equivalent because of the following general fact about sets and equivalence relations. If <math>S</math> is a set, and <math>\sim</math> is an equivalence relation on <math>S</math>, then we can partition <math>S</math> as a disjoint union of ''equivalence classes'' under <math>\! \sim</math>. Two elements <math>a</math> and <math>b</math> are defined to be in the same equivalence class under <math>\! \sim</math> if <math>\! a \sim b</math>.
Conversely, if <math>S</math> is partitioned as a disjoint union of subsets, then the relation of being in the same subset is an equivalence relation on <math>S</math>.
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