> [!NOTE] Definition (Quotient) > Given an [[Equivalence Relation|equivalence relation]] $E$ on a [[Set|set]] $X.$ We denote the set of [[Equivalence Class|equivalence classes]] by $X/E=\{ [x]_{E} \mid x \in X \}$called the *quotient* (*quotient set* or *quotient space*) of $X$ by $E.$ > > [!Example] Examples > - [[Set-theoretic construction of the integers|Integers]]. > - [[Integers Modulo n|Integers moulo n]]. # Properties **Partitions**: The [[Fundamental Theorem on Equivalence Relations|fundamental theorem on equivalence relations]] asserts that the quotient of a set by an equivalence relation is a partition of the set. Moreover, [[For every partition there is a unique equivalence relation|every partition corresponds to a unique equivalence relation and vice versa]]. **Well-defined Operation**: A well-defined binary operation on $X$ is defined as one that maps equivalent elements (representatives of the same equivalence class) to same result. The [[Universal Property of Quotients|universal property of quotients]] asserts that if $f:X\to Y$ is well-defined then we can construct a unique mapping $f_{E}: X/E\to Y$ so $f=f_{E}\circ q_{E}$ where $q_{E}$ is the [[Quotient Map|quotient map]] given by $E.$ # Applications Coset space: ...