> [!NOTE] **Definition** (Well defined function) > Suppose that $X$ and $Y$ are sets. Suppose that $E$ is an [[Equivalence relations|equivalence relation]] $X$. Suppose that $f:X \to Y$ is a [[Function|function]] such that $\text{for all } x, x' \in X, \quad xEx' \implies f(x) = f(x')$then $f$ is *well-defined*. > # Properties 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 $E.$