Well-defined

“Well-defined” is a slightly fuzzy word in math­e­mat­ics. Broadly, an ob­ject is said to be “well-defined” if it has been given a defi­ni­tion that is com­pletely un­am­bigu­ous, can be ex­e­cuted with­out re­gard to any ar­bi­trary choices the math­e­mat­i­cian might make, or gen­er­ally is crisply defined.

(The Wikipe­dia page on well-defined­ness con­tains many ex­am­ples for those who are more com­fortable with math­e­mat­i­cal no­ta­tion.)

Spe­cific instances

Functions

One of the most com­mon uses of the phrase “well-defined” is when talk­ing about func­tions. A func­tion is well-defined if it re­ally is a bona fide func­tion. This usu­ally man­i­fests it­self as the fol­low­ing:

When­ever \(x=y\), we have \(f(x) = f(y)\): that is, the out­put of the func­tion doesn’t de­pend on how we spec­ify the in­put to the func­tion, only on the in­put it­self.

This prop­erty is of­ten pretty easy to check. For in­stance, the func­tion from \(\mathbb{N}\) to it­self given by \(n \mapsto n+1\) is “ob­vi­ously” well-defined: it’s triv­ially ob­vi­ous that if \(n=m\) then \(f(n) = f(m)\).

How­ever, some­times it is not so easy. The func­tion \(\mathbb{N} \to \mathbb{N}\) given by “take the num­ber of prime fac­tors” is not ob­vi­ously well-defined, be­cause it could in prin­ci­ple be the case that some num­ber \(n\) is equal to both \(p_1 p_2 p_3\) and \(q_1 q_2\) for some primes \(p_1, p_2, p_3, q_1, q_2\); then our pu­ta­tive func­tion might plau­si­bly at­tempt to out­put ei­ther \(3\) or \(2\) on the same nat­u­ral num­ber in­put \(n\), so the func­tion would not be well-defined. (It turns out that there is a non-triv­ial the­o­rem, the Fun­da­men­tal The­o­rem of Arith­metic, guaran­tee­ing that this func­tion is in fact well-defined.)

Well-defined­ness in this con­text comes up very of­ten when we are at­tempt­ing to take a quo­tient. The fact that we can take the quo­tient of a set \(X\) by an equiv­alence re­la­tion \(\sim\) is tan­ta­mount to say­ing:

The func­tion \(X \to \frac{X}{\sim}\), given by \(x \mapsto [x]\) the equiv­alence class of \(X\), is well-defined.


Another, differ­ent, way a func­tion could fail to be well-defined is if we tried to take the func­tion \(\mathbb{N} \to \mathbb{N}\) given by \(n \mapsto n-5\). This func­tion is un­am­bigu­ous, but it’s not well-defined, be­cause on the in­put \(2\) it tries to out­put \(-3\), which is not in the speci­fied codomain.

Parents:

  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.