Editing Set-builder notation (section)

== Sets defined by a predicate ==
Set-builder notation can be used to describe a set that is defined by a [[predicate (mathematical logic)|predicate]], that is, a logical formula that evaluates to ''true'' for an element of the set, and ''false'' otherwise.<ref>Michael J Cullinan, 2012, ''A Transition to Mathematics with Proofs'', Jones & Bartlett, pp.&nbsp;44ff.</ref> In this form, set-builder notation has three parts: a variable, a [[Colon (punctuation)|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:
:<math>\{x \mid \Phi(x)\}</math>
or
:<math>\{x : \Phi(x)\}.</math>
The vertical bar (or colon) is a separator that can be read as "'''such that'''", <!--bold, because "such that" redirects here--> "for which", or "with the property that".  The formula {{math|&Phi;(''x'')}} is said to be the ''rule'' or the ''predicate''. All values of {{math|''x''}} for which the predicate holds (is true) belong to the set being defined. All values of {{math|''x''}} for which the predicate does not hold do not belong to the set.  Thus <math>\{x \mid \Phi(x)\}</math> is the set of all values of {{math|''x''}} that satisfy the formula {{math|&Phi;}}.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Set|url=https://mathworld.wolfram.com/Set.html|access-date=2020-08-20|website=mathworld.wolfram.com|language=en}}</ref> It may be the [[empty set]], if no value of {{math|''x''}} satisfies the formula.

=== Specifying the domain ===

A domain {{math|''E''}} can appear on the left of the vertical bar:<ref>{{Cite web|title=Set-Builder Notation|url=https://www.mathsisfun.com/sets/set-builder-notation.html|access-date=2020-08-20|website=mathsisfun.com}}</ref> 
:<math>\{x \in E  \mid  \Phi(x)\},</math> 
or by adjoining it to the predicate:
:<math>\{ x \mid x \in E \text{ and } \Phi(x)\}\quad\text{or}\quad\{ x \mid x \in E \land \Phi(x)\}.</math> 
The &isin; symbol here denotes [[set membership]], while the <math>\land</math> symbol denotes the logical "and" operator, known as [[logical conjunction]]. This notation represents the set of all values of {{math|''x''}} that belong to some given set {{math|''E''}} for which the predicate is true (see "[[#Set existence axiom|Set existence axiom]]" below). If <math>\Phi(x)</math> is a conjunction <math>\Phi_1(x)\land\Phi_2(x)</math>, then <math>\{x \in E  \mid  \Phi(x)\}</math> is sometimes written <math>\{x \in E  \mid  \Phi_1(x), \Phi_2(x)\}</math>, using a comma instead of the symbol <math>\land</math>.

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 <math>\{x ~|~ x\not\in x\},</math> although seemingly well formed as a set builder expression, cannot define a set without producing a contradiction.<ref>{{cite encyclopedia | first1=Andrew David | last1=Irvine | first2=Harry | last2=Deutsch | orig-year=1995 | date=9 October 2016 | url=http://plato.stanford.edu/entries/russell-paradox/ | title=Russell's Paradox | encyclopedia=Stanford Encyclopedia of Philosophy | access-date=6 August 2017}}</ref>

In cases where the set {{math|''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. 
* <math>\{ x \in \mathbb{R}   \mid  x > 0\}</math> is the set of all strictly [[positive number|positive]] [[real number]]s, which can be written in [[interval notation]] as <math>(0, \infty)</math>.
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* <math>\{ x \in \mathbb{R}  \mid |x| = 1 \}</math> is the set <math>\{-1, 1\}</math>. This set can also be defined as <math>\{x \in \mathbb{R} \mid x^2 = 1\}</math>; see [[#Equivalent predicates yield equal sets|equivalent predicates yield equal sets]] below.
* For each integer {{math|''m''}}, we can define <math>G_m = \{x  \in \mathbb{Z}  \mid  x \ge m \} = \{ m, m + 1, m + 2, \ldots\}</math>. As an example, <math>G_3 = \{x \in \mathbb{Z} \mid x \ge 3 \} = \{ 3, 4, 5, \ldots\}</math> and <math>G_{-2} = \{ -2, -1, 0, \ldots\}</math>.
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* <math>\{ (x,y) \in \mathbb{R} \times \mathbb{R} \mid 0 < y < f(x) \}</math>  is the set of pairs of real numbers such that ''y'' is greater than 0 and less than {{math|''f''(''x'')}}, for a given [[function (mathematics)|function]] {{math|''f''}}.   Here the [[cartesian product]] <math>\mathbb{R}\times\mathbb{R}</math> denotes the set of ordered pairs of real numbers. 
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* <math>\{n   \in \mathbb{N} \mid  (\exists k)  [k\in \mathbb{N}\land n = 2k]  \} </math> is the set of all [[even number|even]] [[natural number]]s. The <math>\land</math> sign stands for "and", which is known as [[logical conjunction]]. The &exist; sign stands for "there exists", which is known as [[existential quantification]]. So for example, <math> (\exists x) P(x)</math> is read as  "there exists an {{math|''x''}} such that {{math|''P''(''x'')}}".
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*  <math>\{n \mid  (\exists k \in \mathbb{N} )  [n = 2k]  \} </math> is a notational variant for the same set of even natural numbers. It is not necessary to specify that {{math|''n''}} is a natural number, as this is implied by the formula on the right.
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* <math>\{a \in \mathbb{R} \mid (\exists p\in \mathbb{Z} )(\exists q\in \mathbb{Z} )[ q \not = 0 \land aq=p] \}</math> is the set of [[rational number]]s; that is, real numbers that can be written as the ratio of two [[integer]]s.