Set-builder notation

A Venn diagram illustrating the intersection of two sets

Mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.

- Set-builder notation

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Zermelo–Fraenkel set theory

Axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.

A Venn diagram illustrating the intersection of two sets

Subsets are commonly constructed using set builder notation.

List comprehension

Syntactic construct available in some programming languages for creating a list based on existing lists.

Syntax highlighting and indent style are often used to aid programmers in recognizing elements of source code. This Python code uses color coded highlighting.

It follows the form of the mathematical set-builder notation (set comprehension) as distinct from the use of map and filter functions.

Primitive notion

Concept that is not defined in terms of previously-defined concepts.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

To establish sets he also requires propositional functions as primitive, as well as the phrase "such that" as used in set builder notation.

Notation

System of graphics or symbols, characters and abbreviated expressions, used in artistic and scientific disciplines to represent technical facts and quantities by convention.

Color-coding hot- and cold-water faucets (taps) is common in many cultures but, as this example shows, the coding may be rendered meaningless because of context. The two faucets (taps) probably were sold as a coded set, but the code is unusable (and ignored), as there is a single water supply.

Set-builder notation, a formal notation for defining sets in set theory

Riemann integral

The first rigorous definition of the integral of a function on an interval.

The integral as the area of a region under a curve.
A sequence of Riemann sums over a regular partition of an interval. The number on top is the total area of the rectangles, which converges to the integral of the function.
The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero.

This region can be expressed in set-builder notation as

Willard Van Orman Quine

American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century".

He was delighted to discover early in his career that all of first order logic and set theory could be grounded in a mere two primitive notions: abstraction and inclusion.

Type theory

Formal system in which every "term" has a "type".

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Set theory has set-builder notation. It can create any set that can be defined. This allows it to create Uncountable sets. Type theories are syntactic, which limits them to a countably infinite terms. Additionally, most type theories require computation to always halt and limit themselves to recursively generable terms. As a result, most type theories do not use the Real numbers but the Computable numbers.

Big O notation

Mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity.

Example of Big O notation:.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations N versus input size n for each function

For these reasons, it would be more precise to use set notation and write f(x) ∈ O(g(x)) (read as: "f(x) is an element of O(g(x))", or "f(x) is in the set O(g(x))"), thinking of O(g(x)) as the class of all functions h(x) such that |h(x)| ≤ C|g(x)| for some constant C.

Interval (mathematics)

Set of real numbers that contains all real numbers lying between any two numbers of the set.

The addition x + a on the number line. All numbers greater than x and less than x + a fall within that open interval.

Thus, in set builder notation,

Colon (punctuation)

Punctuation mark consisting of two equally sized dots placed on the same vertical line.

15th century Bible text in Ge'ez script showing colons between the words.

In mathematical logic, when using set-builder notation for describing the characterizing property of a set, it is used as an alternative to a vertical bar (which is the ISO 31-11 standard), to mean "such that".