Editing Set-builder notation (section)

=== 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.