Editing Set (mathematics) (section)

===Set-builder notation===
{{main|Set-builder notation}}

Set-builder notation specifies a set as being the set of all elements that satisfy some [[logical formula]].<ref name="Ruda2011">{{cite book|author=Frank Ruda|title=Hegel's Rabble: An Investigation into Hegel's Philosophy of Right|url=https://books.google.com/books?id=VV0SBwAAQBAJ&pg=PA151|date=6 October 2011|publisher=Bloomsbury Publishing|isbn=978-1-4411-7413-0|page=151}}</ref><ref name="Lucas1990">{{cite book|author=John F. Lucas|title=Introduction to Abstract Mathematics|url=https://books.google.com/books?id=jklsb5JUgoQC&pg=PA108|year=1990|publisher=Rowman & Littlefield|isbn=978-0-912675-73-2|page=108}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Set|url=https://mathworld.wolfram.com/Set.html|access-date=2020-08-19|website=Wolfram MathWorld |language=en}}</ref> More precisely, if {{tmath|P(x)}} is a logical formula depending on a [[variable (mathematics)|variable]] {{tmath|x}}, which evaluates to ''true'' or ''false'' depending on the value of {{tmath|x}}, then 
<math display=block>\{x \mid P(x)\}</math>
or<ref name="Steinlage1987">{{cite book|author=Ralph C. Steinlage|title=College Algebra|url=https://books.google.com/books?id=lcg3gY3444IC|year=1987|publisher=West Publishing Company|isbn=978-0-314-29531-6}}</ref>
<math display=block>\{x : P(x)\}</math>
denotes the set of all {{tmath|x}} for which {{tmath|P(x)}} is true.<ref name=":0" /> For example, a set {{mvar|F}} can be specified as follows:
<math display="block">F = \{n \mid n \text{ is an integer, and } 0 \leq n \leq 19\}.</math>
In this notation, the [[vertical bar]] "|" is read as "such that", and the whole formula can be read as "{{mvar|F}} is the set of all  {{mvar|n}} such that {{mvar|n}} is an integer in the range from 0 to 19 inclusive".

Some logical formulas, such as {{tmath| \color{red}{S \text{ is a set} } }} or {{tmath|\color{red}{S \text{ is a set and } S\not\in S} }} cannot be used in set-builder notation because there is no set for which the elements are characterized by the formula. There are several ways for avoiding the problem. One may prove that the formula defines a set; this is often almost immediate, but may be very difficult. 

One may also introduce a larger set {{tmath|U}} that must contain all elements of the specified set, and write the notation as
<math display=block>\{x\mid x\in U \text{ and ...}\}</math>
or
<math display=block>\{x\in U\mid \text{ ...}\}.</math>

One may also define {{tmath|U}} once for all and take the convention that every variable that appears on the left of the vertical bar of the notation represents an element of {{tmath|U}}. This amounts to say that {{tmath|x\in U}} is implicit in set-builder notation. In this case, {{tmath|U}} is often called ''the [[domain of discourse]]'' or a ''[[universe (mathematics)|universe]]''.

For example, with the convention that a lower case Latin letter may represent a [[real number]] and nothing else, the [[expression (mathematics)|expression]]
<math display=block>\{x\mid x\not\in \Q\}</math>
is an abbreviation of 
<math display="block">\{x\in \R \mid x\not\in \Q\},</math>
which defines the [[irrational number]]s.