# well-defined

A mathematical concept is *well-defined* (German wohldefiniert, French bien défini), if its contents is
on the form or the alternative representative which is used for defining it.

For example, in defining the http://planetmath.org/FractionPowerpower ${x}^{r}$ with $x$ a positive real and $r$ a rational number^{},
we can freely choose the fraction form $\frac{m}{n}$ ($m\in \mathbb{Z}$, $n\in {\mathbb{Z}}_{+}$) of $r$ and take

$${x}^{r}:=\sqrt[n]{{x}^{m}}$$ |

and be sure that the value of ${x}^{r}$ does not depend on that choice (this is justified in the entry fraction power). So, the ${x}^{r}$ is well-defined.

In many instances well-defined is a synonym for the formal definition of a function between sets. For example, the function $f(x):={x}^{2}$ is a well-defined function from the real numbers to the real numbers because every input, $x$, is assigned to precisely one output, ${x}^{2}$. However, $f(x):=\pm \sqrt{x}$ is not well-defined in that one input $x$ can be assigned any one of two possible outputs, $\sqrt{x}$ or $-\sqrt{x}$.

More subtle examples include expressions such as

$$f\left(\frac{a}{b}\right):=a+b,\frac{a}{b}\in \mathbb{Q}.$$ |

Certainly every input has an output, for instance, $f(1/2)=3$. However, the expression is *not*
well-defined since $1/2=2/4$ yet $f(1/2)=3$ while $f(2/4)=6$ and $3\ne 6$.

One must question whether a function is well-defined whenever it is defined on a domain of equivalence classes^{}
in such a manner that each output is determined for a representative of each equivalence class. For example, the
function $f(a/b):=a+b$ was defined using the representative $a/b$ of the equivalence class of fractions
equivalent^{} to $a/b$.

Title | well-defined |
---|---|

Canonical name | Welldefined |

Date of creation | 2013-03-22 17:31:32 |

Last modified on | 2013-03-22 17:31:32 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 00A05 |

Synonym | well defined |

Related topic | function |

Related topic | WellDefinednessOfProductOfFinitelyGeneratedIdeals |