**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Well-defined expression

Summary

In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill defined or ambiguous. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if takes real numbers as input, and if does not equal then is not well defined (and thus not a function). The term well defined can also be used to indicate that a logical expression is unambiguous or uncontradictory.
A function that is not well defined is not the same as a function that is undefined. For example, if , then even though is undefined does not mean that the function is not well defined – but simply that 0 is not in the domain of .
Let be sets, let and "define" as if and if .
Then is well defined if . For example, if and , then would be well defined and equal to .
However, if , then would not be well defined because is "ambiguous" for . For example, if and , then would have to be both 0 and 1, which makes it ambiguous. As a result, the latter is not well defined and thus not a function.
In order to avoid the quotation marks around "define" in the previous simple example, the "definition" of could be broken down into two simple logical steps:
While the definition in step 1 is formulated with the freedom of any definition and is certainly effective (without the need to classify it as "well defined"), the assertion in step 2 has to be proved. That is, is a function if and only if , in which case – as a function – is well defined.
On the other hand, if , then for an , we would have that and , which makes the binary relation not functional (as defined in Binary relation#Special types of binary relations) and thus not well defined as a function. Colloquially, the "function" is also called ambiguous at point (although there is per definitionem never an "ambiguous function"), and the original "definition" is pointless.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (9)

Related people (1)

Related concepts (6)

Related courses (11)

Related lectures (37)

Modulo

In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the modulus of the operation). Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. For example, the expression "5 mod 2" would evaluate to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3.

Fraction

A fraction (from fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: and ) consists of an integer numerator, displayed above a line (or before a slash like ), and a non-zero integer denominator, displayed below (or after) that line.

Division by zero

In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as , where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number that, when multiplied by 0, gives a (assuming ); thus, division by zero is undefined (a type of singularity). Since any number multiplied by zero is zero, the expression is also undefined; when it is the form of a limit, it is an indeterminate form.

MATH-106(f): Analysis II

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs
variables.

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-225: Topology

On étudie des notions de topologie générale: unions et quotients d'espaces topologiques; on approfondit les notions de revêtements et de groupe fondamental,et d'attachements de cellules et on démontre

Partons and Hadrons: Strong Force and Deep Inelastic Scattering

Explores partons, hadrons, strong force, deep inelastic scattering, elastic and inelastic scattering, and Bjorken scaling.

Linear Transformations: Kernels and Images

Covers kernels and images of linear transformations between vector spaces, illustrating properties and providing proofs.

Advanced Analysis II: Differential Equations

Covers the review of differential equations and Lipschitz conditions for solving them.

Eva Bayer Fluckiger, Daniel Arnold Moldovan

We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be unimodular or defined o ...

The calculation of the inverse scalelengths $R/L_T$, $R/L_n$ may depend on the choice of the radial variable, depending on the definition of the scalelength. The value of $\lambda_T$ and $\lambda_n$, as defined in [O. Sauter {\it et al}, Phys. Plasmas {\bf ...

Introduced 50 years ago by David Kazhdan, Kazhdan's Property (T) has quickly become an active research area in mathematics, with a lot of important results. A few years later, this property has been generalized to discrete group actions by Robert J. Zimmer ...