Editing Set theory (section)

==Basic concepts and notation==
{{Main|Set (mathematics)|Algebra of sets}}

Set theory begins with a fundamental [[binary relation]] between an object {{mvar|o}} and a set {{mvar|A}}. If {{mvar|o}} is a ''[[set membership|member]]'' (or ''element'') of {{mvar|A}}, the notation {{math|''o'' ∈ ''A''}} is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }.<ref>{{Cite web|title=Introduction to Sets|url=https://www.mathsisfun.com/sets/sets-introduction.html|access-date=2020-08-20|website=www.mathsisfun.com|archive-date=2006-07-16|archive-url=https://web.archive.org/web/20060716000900/https://www.mathsisfun.com/sets/sets-introduction.html|url-status=live}}</ref> Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets.

A derived binary relation between two sets is the subset relation, also called ''set inclusion''. If all the members of set {{mvar|A}} are also members of set {{mvar|B}}, then {{mvar|A}} is a ''[[subset]]'' of {{mvar|B}}, denoted {{math|''A'' ⊆ ''B''}}. For example, {{math|{{mset|1, 2}}}} is a subset of {{math|{{mset|1, 2, 3}}}}, and so is {{math|{{mset|2}}}} but {{math|{{mset|1, 4}}}} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term ''[[proper subset]]'' is defined, variously denoted <math>A\subset B</math>, <math>A\subsetneq B</math>, or <math>A\subsetneqq B</math> (note however that the notation <math>A\subset B</math> is sometimes used synonymously with <math>A\subseteq B</math>; that is, allowing the possibility that {{mvar|A}} and {{mvar|B}} are equal).  We call {{mvar|A}} a ''proper subset'' of {{mvar|B}} if and only if {{mvar|A}} is a subset of {{mvar|B}}, but {{mvar|A}} is not equal to {{mvar|B}}. Also, 1, 2, and 3 are members (elements) of the set {{math|{{mset|1, 2, 3}}}}, but are not subsets of it; and in turn, the subsets, such as {{math|{{mset|1}}}}, are not members of the set {{math|{{mset|1, 2, 3}}}}. More complicated relations can exist; for example, the set {{math|{{mset|1}}}} is both a member and a proper subset of the set {{math|{{mset|1, {{mset|1}}}}}}. 

Just as [[arithmetic]] features [[binary operation]]s on [[number]]s, set theory features binary operations on sets.<ref>{{citation|url=https://archive.org/details/introductoryreal00kolm_0/page/2|title=Introductory Real Analysis|last1=Kolmogorov|first1=A.N.|last2=Fomin|first2=S.V.|publisher=Dover Publications|year=1970|isbn=0486612260|edition=Rev. English|location=New York|pages=[https://archive.org/details/introductoryreal00kolm_0/page/2 2–3]|oclc=1527264|author-link=Andrey Kolmogorov|author-link2=Sergei Fomin|url-access=registration}}</ref> The following is a partial list of them:
*''[[Union (set theory)|Union]]'' of the sets {{mvar|A}} and {{mvar|B}}, denoted {{math|''A'' ∪ ''B''}}, is the set of all objects that are a member of {{mvar|A}}, or {{mvar|B}}, or both.<ref>{{Cite web|title=set theory {{!}} Basics, Examples, & Formulas|url=https://www.britannica.com/science/set-theory|access-date=2020-08-20|website=Encyclopedia Britannica|language=en|archive-date=2020-08-20|archive-url=https://web.archive.org/web/20200820100726/https://www.britannica.com/science/set-theory|url-status=live}}</ref> For example, the union of {{math|{{mset|1, 2, 3}}}} and {{math|{{mset|2, 3, 4}}}} is the set {{math|{{mset|1, 2, 3, 4}}}}.
*''[[Intersection (set theory)|Intersection]]'' of the sets {{mvar|A}} and {{mvar|B}}, denoted {{math|''A'' ∩ ''B''}}, is the set of all objects that are members of both {{mvar|A}} and {{mvar|B}}.<ref>{{cite book |last1=Kaplansky |first1=Irving |editor1-last=De Prima |editor1-first=Charles |title=Set Theory and Metric Spaces |date=1972 |publisher=Allyn and Bacon |location=Boston |page=4 |language=en}}</ref> For example, the intersection of {{math|{{mset|1, 2, 3}}}} and {{math|{{mset|2, 3, 4}}}} is the set {{math|{{mset|2, 3}}}}.
*''[[Set difference]]'' of {{mvar|U}} and {{mvar|A}}, denoted {{math|''U'' &setminus; ''A''}}, is the set of all members of {{mvar|U}} that are not members of {{mvar|A}}.<ref>{{cite book |last1=Kaplansky |first1=Irving |editor1-last=De Prima |editor1-first=Charles |title=Set Theory and Metric Spaces |date=1972 |publisher=Allyn and Bacon |location=Boston |page=5–6 |language=en}}</ref>  The set difference {{math|{1, 2, 3} &setminus; {2, 3, 4} }} is {{math|{{mset|1}}}}, while conversely, the set difference {{math|{2, 3, 4} &setminus; {{mset|1, 2, 3}}}} is {{math|{{mset|4}}}}. When {{mvar|A}} is a subset of {{mvar|U}}, the set difference {{math|''U'' &setminus; ''A''}} is also called the ''[[complement (set theory)|complement]]'' of {{mvar|A}} in {{mvar|U}}. In this case, if the choice of {{mvar|U}} is clear from the context, the notation {{math|''A''<sup>''c''</sup>}} is sometimes used instead of {{math|''U'' &setminus; ''A''}}, particularly if {{mvar|U}} is a [[universal set]] as in the study of [[Venn diagram]]s.<ref>{{cite book |last1=Kaplansky |first1=Irving |editor1-last=De Prima |editor1-first=Charles |title=Set Theory and Metric Spaces |date=1972 |publisher=Allyn and Bacon |location=Boston |page=5–6 |language=en}}</ref>
*''[[Symmetric difference]]'' of sets {{mvar|A}} and {{mvar|B}}, denoted {{math|''A'' △ ''B''}} or {{math|''A'' ⊖ ''B''}}, is the set of all objects that are a member of exactly one of {{mvar|A}} and {{mvar|B}} (elements which are in one of the sets, but not in both). For instance, for the sets {{math|{{mset|1, 2, 3}}}} and {{math|{{mset|2, 3, 4}}}}, the symmetric difference set is {{math|{{mset|1, 4}}}}. It is the set difference of the union and the intersection, {{math|(''A'' ∪ ''B'') &setminus; (''A'' ∩ ''B'')}} or {{math|(''A'' &setminus; ''B'') ∪ (''B'' &setminus; ''A'')}}.
*''[[Cartesian product]]'' of {{mvar|A}} and {{mvar|B}}, denoted {{math|''A'' × ''B''}}, is the set whose members are all possible [[ordered pair]]s {{math|(''a'', ''b'')}}, where {{mvar|a}} is a member of {{mvar|A}} and {{mvar|b}} is a member of {{mvar|B}}. For example, the Cartesian product of {1, 2} and {red, white} is {{nowrap|1={(1, red), (1, white), (2, red), (2, white)}.}}<ref>{{cite book |last1=Kaplansky |first1=Irving |editor1-last=De Prima |editor1-first=Charles |title=Set Theory and Metric Spaces |date=1972 |publisher=Allyn and Bacon |location=Boston |page=19 |language=en}}</ref>

Some basic sets of central importance are the set of [[natural number]]s, the set of [[real number]]s and the [[empty set]] – the unique set containing no elements. The empty set is also occasionally called the ''null set'',<ref>{{Citation|last=Bagaria|first=Joan|title=Set Theory|date=2020|url=https://plato.stanford.edu/archives/spr2020/entries/set-theory/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Spring 2020|publisher=Metaphysics Research Lab, Stanford University|access-date=2020-08-20}}</ref> though this name is ambiguous and can lead to several interpretations. The empty set can be denoted with empty braces "<math> \{ \} </math>" or the symbol "<math> \varnothing </math>" or "<math> \emptyset </math>".

The [[power set]] of a set {{mvar|A}}, denoted <math>\mathcal{P}(A)</math>, is the set whose members are all of the possible subsets of {{mvar|A}}. For example, the power set of {{math|{{mset|1, 2}}}} is {{math|{{mset| {{mset}}, {{mset|1}}, {{mset|2}}, {{mset|1, 2}} }}}}. Notably, <math>\mathcal{P}(A)</math> contains both {{mvar|A}} and the empty set.