In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for defining a set by stating the properties that its members must satisfy.

Set-builder notation is be used to define a set using a predicate, that is, a logical formula that evaluates to true for each element contained in the set, and false for those elements not contained in the set.[1][2] In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the separator, and a rule on the right of it. These three parts are contained in curly brackets:

or

The vertical bar (or colon) is a separator that can be read as "such that", "for which", or "with the property that". The formula Φ(x) is said to be the rule or the predicate. All values of x for which the predicate holds (is true) belong to the set being defined. All values of x for which the predicate does not hold do not belong to the set. Thus is the set of all values of x that satisfy the formula Φ.[3] It may be the empty set, if no value of x satisfies the formula.

Specifying the domain

A domain E can appear on the left of the vertical bar:[4]

or by adjoining it to the predicate:

The ∈ symbol here denotes set membership, while the symbol denotes the logical "and" operator, known as logical conjunction. This notation represents the set of all values of x that belong to some given set E for which the predicate is true (see "Set existence axiom" below). If is a conjunction , then is sometimes written , using a comma instead of the symbol .

In general, it is not a good idea to consider sets without defining a domain of discourse, as this would represent the subset of all possible things that may exist for which the predicate is true. This can easily lead to contradictions and paradoxes. For example, Russell's paradox shows that the expression although seemingly well formed as a set builder expression, cannot define a set without producing a contradiction.[5]

In cases where the set E is clear from context, it may be not explicitly specified. It is common in the literature for an author to state the domain ahead of time, and then not specify it in the set-builder notation. For example, an author may say something such as, "Unless otherwise stated, variables are to be taken to be natural numbers," though in less formal contexts where the domain can be assumed, a written mention is often unnecessary.

Examples

The following examples illustrate particular sets defined by set-builder notation via predicates. In each case, the domain is specified on the left side of the vertical bar, while the rule is specified on the right side.

More complex expressions on the left side of the notation

An extension of set-builder notation replaces the single variable x with an expression. So instead of , we may have which should be read

.

For example:

When inverse functions can be explicitly stated, the expression on the left can be eliminated through simple substitution. Consider the example set . Make the substitution , which is to say , then replace t in the set builder notation to find

Equivalent predicates yield equal sets

Two sets are equal if and only if they have the same elements. Sets defined by set builder notation are equal if and only if their set builder rules, including the domain specifiers, are equivalent. That is

if and only if

.

Therefore, in order to prove the equality of two sets defined by set builder notation, it suffices to prove the equivalence of their predicates, including the domain qualifiers.

For example,

because the two rule predicates are logically equivalent:

This equivalence holds because, for any real number x, we have if and only if x is a rational number with . In particular, both sets are equal to the set .

Set existence axiom

In many formal set theories, such as Zermelo–Fraenkel set theory, set builder notation is not part of the formal syntax of the theory. Instead, there is a set existence axiom scheme, which states that if E is a set and Φ(x) is a formula in the language of set theory, then there is a set Y whose members are exactly the elements of E that satisfy Φ:

The set Y obtained from this axiom is exactly the set described in set builder notation as .

In programming languages

Main article: List comprehension

A similar notation available in a number of programming languages (notably Python and Haskell) is the list comprehension, which combines map and filter operations over one or more lists.

It has been suggested that parts of this page be moved into List comprehension. (Discuss) (December 2023)

In Python, the set-builder's braces are replaced with square brackets, parentheses, or curly braces, giving list, generator, and set objects, respectively. Python uses an English-based syntax. Haskell replaces the set-builder's braces with square brackets and uses symbols, including the standard set-builder vertical bar.

The same can be achieved in Scala using Sequence Comprehensions, where the "for" keyword returns a list of the yielded variables using the "yield" keyword.[6]

Consider these set-builder notation examples in some programming languages:

Example 1 Example 2
Set-builder
Python
{l for l in L}
{(k, x) for k in K for x in X if P(x)}
Haskell
[l | l <- ls]
[(k, x) | k <- ks, x <- xs, p x]
Scala
for (l <- L) yield l
for (k <- K; x <- X if P(x)) yield (k,x)
C#
from l in L select l
from k in K from x in X where P(x) select (k,x)
SQL
SELECT l FROM L_set
 SELECT k, x FROM K_set, X_set WHERE P(x)
Prolog
setof(L,member(L,Ls),Result)
setof((K,X),(member(K,Ks),member(X,Xs),call(P,X)),Result)
Erlang
[l || l <- ls]
Julia
[l for l  L]
[(k, x) for k  K for x  X if P(x)]

The set builder notation and list comprehension notation are both instances of a more general notation known as monad comprehensions, which permits map/filter-like operations over any monad with a zero element.

See also

Notes

  1. ^ Rosen, Kenneth H. (2012). Discrete Mathematics and its Applications (7th ed.). New York, NY: McGraw-Hill. ISBN 978-0-07-338309-5.
  2. ^ Cullinan, Michael J. (2012). A Transition to Mathematics with Proofs. Jones & Bartlett. ISBN 978-1-44-962778-2.
  3. ^ Weisstein, Eric W. "Set". mathworld.wolfram.com. Retrieved 20 August 2020.
  4. ^ "Set-Builder Notation". mathsisfun.com. Retrieved 20 August 2020.
  5. ^ Irvine, Andrew David; Deutsch, Harry (9 October 2016) [1995]. "Russell's Paradox". Stanford Encyclopedia of Philosophy. Retrieved 6 August 2017.
  6. ^ "Sequence Comprehensions". Scala. Retrieved 6 August 2017.