Sets:Definitions - Revision history

Khanh: /* Power sets */

2022-02-04T20:03:30Z

Power sets

Khanh: /* Cardinality */

2022-02-04T19:57:52Z

Cardinality

Khanh: /* Special sets of numbers in mathematics */

2022-02-04T19:53:30Z

Special sets of numbers in mathematics

Khanh: /* Euler and Venn diagrams */

2022-02-04T19:50:16Z

Euler and Venn diagrams

Khanh: /* Subsets */

2022-02-04T19:49:09Z

Subsets

Khanh: /* Singleton sets */

2022-02-04T19:43:38Z

Singleton sets

Khanh: /* The empty set */

2022-02-04T19:40:29Z

The empty set

Khanh: /* Membership */

2022-02-04T19:39:18Z

Membership

Khanh: /* How sets are defined and set notation */

2022-02-04T19:32:53Z

How sets are defined and set notation

Khanh: /* Naïve set theory */

2022-02-04T19:30:08Z

Naïve set theory

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 ==Power sets==
-The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{{mset|1, 2, 3}}}} is {{math|{{mset|∅, {{mset|1}}, {{mset|2}}, {{mset|3}}, {{mset|1, 2}}, {{mset|1, 3}}, {{mset|2, 3}}, {{mset|1, 2, 3}}}}}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or {{math|2{{sup|''S''}}}}.
+The power set of a set {{math|''S''}} is the set of all subsets of {{math|''S''}}. The empty set and {{math|''S''}} itself are elements of the power set of {{math|''S''}}, because these are both subsets of {{math|''S''}}. For example, the power set of {{math|{1, 2, 3}}} is
-If {{math|''S''}} has {{math|''n''}} elements, then {{math|''P''(''S'')}} has {{math|2{{sup|''n''}}}} elements. For example, {{math|{{mset|1, 2, 3}}}} has three elements, and its power set has {{math|2{{sup|3}} {{=}} 8}} elements, as shown above.
+{∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. The power set of a set {{math|''S''}} is commonly written as {{math|''P''(''S'')}} or 2<sup>''S''</sup>.
 If {{math|''S''}} is infinite (whether countable or uncountable), then {{math|''P''(''S'')}} is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of {{math|''S''}} with the elements of {{math|''P''(''S'')}} will leave some elements of {{math|''P''(''S'')}} unpaired. (There is never a bijection from {{math|''S''}} onto {{math|''P''(''S'')}}.)

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 ==Cardinality==
-The cardinality of a set {{math|''S''}}, denoted {{math|{{mabs|''S''}}}}, is the number of members of {{math|''S''}}. For example, if {{math|''B'' {{=}} {{mset|blue, white, red}}}}, then {{math|1={{mabs|B}} = 3}}. Repeated members in roster notation are not counted, so {{math|1={{mabs|{{mset|blue, white, red, blue, white}}}} = 3}}, too.
+The cardinality of a set {{math|''S''}}, denoted |''S''|, is the number of members of {{math|''S''}}. For example, if ''B'' = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted, so
 More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

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 * <math>\bold N</math> or <math>\mathbb N</math>, the set of all natural numbers: <math>\bold N=\{0,1,2,3,...\}</math> (often, authors exclude {{math|0}});
 * <math>\bold Z</math> or <math>\mathbb Z</math>, the set of all integers (whether positive, negative or zero): <math>\bold Z=\{...,-2,-1,0,1,2,3,...\}</math>;
-* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, {{math|&minus;{{sfrac|7|4}} ∈ '''Q'''}} and {{math|5 {{=}} {{sfrac|5|1}} ∈ '''Q'''}};
+* <math>\bold Q</math> or <math>\mathbb Q</math>, the set of all rational numbers (that is, the set of all proper and improper fractions): <math>\bold Q=\left\{\frac {a}{b}\mid a,b\in\bold Z,b\ne0\right\}</math>. For example, <math>-\tfrac{7}{4}</math> ∈ '''Q''' and <math>5 = \tfrac{5}{1}</math> ∈ '''Q''';
 * <math>\bold R</math> or <math>\mathbb R</math>, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as <math>\sqrt2</math> that cannot be rewritten as fractions, as well as transcendental numbers such as {{mvar|π}} and {{math|''e''}});
-* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: {{math|1='''C''' = {{mset|''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}}}}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.
+* <math>\bold C</math> or <math>\mathbb C</math>, the set of all complex numbers: '''C''' = {''a'' + ''bi'' | ''a'', ''b'' ∈ '''R'''}, for example, {{math|1 + 2''i'' ∈ '''C'''}}.
 Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

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 An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If {{mvar|A}} is a subset of {{mvar|B}}, then the region representing {{mvar|A}} is completely inside the region representing {{mvar|B}}. If two sets have no elements in common, the regions do not overlap.
-A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into {{math|2{{sup|''n''}}}} zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).
+A Venn diagram, in contrast, is a graphical representation of {{mvar|n}} sets in which the {{mvar|n}} loops divide the plane into 2<sup>''n''</sup> zones such that for each way of selecting some of the {{mvar|n}} sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}, there should be a zone for the elements that are inside {{mvar|A}} and {{mvar|C}} and outside {{mvar|B}} (even if such elements do not exist).
 ==Special sets of numbers in mathematics==

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 Examples:
 * The set of all humans is a proper subset of the set of all mammals.
-* {{mset|1, 3}} ⊂ {{mset|1, 2, 3, 4}}.
+* {1, 3} ⊂ {1, 2, 3, 4}.
-* {{mset|1, 2, 3, 4}} ⊆ {{mset|1, 2, 3, 4}}.
+* {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
 The empty set is a subset of every set, and every set is a subset of itself:

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 ==Singleton sets==
-A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {{mset|''x''}}, where ''x'' is the element.
+A ''singleton set'' is a set with exactly one element; such a set may also be called a ''unit set''. Any such set can be written as {''x''}, where ''x'' is the element.
-The set {{mset|''x''}} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.
+The set {''x''} and the element ''x'' mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.
 ==Subsets==

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 ==The empty set==
-The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or {{mset|&nbsp;}} or {{math|ϕ}} (or {{mvar|ϕ}}).
+The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted {{math|∅}} or <math>\emptyset</math> or {&nbsp;} or {{math|ϕ}} (or {{mvar|ϕ}}).
 ==Singleton sets==

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 If {{mvar|B}} is a set and {{mvar|x}} is an element of {{mvar|B}}, this is written in shorthand as {{math|''x'' ∈ ''B''}}, which can also be read as "''x'' belongs to ''B''", or "''x'' is in ''B''". The statement "''y'' is not an element of ''B''" is written as {{math|''y'' ∉ ''B''}}, which can also be read as or "''y'' is not in ''B''".
-For example, with respect to the sets {{math|1=''A'' = {{mset|1, 2, 3, 4}}}}, {{math|1=''B'' = {{mset|blue, white, red}}}}, and {{math|1=''F'' = {{mset|''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19}}}},
+For example, with respect to the sets {{math|1=''A'' = {1, 2, 3, 4}}}, {{math|1=''B'' = {blue, white, red}}}, and ''F'' = {''n'' | ''n'' is an integer, and 0 ≤ ''n'' ≤ 19},
 :{{math|4 ∈ ''A''}} and {{math|12 ∈ ''F''}}; and
 :{{math|20 ∉ ''F''}} and {{math|green ∉ ''B''}}.

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 ===Roster notation===
 '''Roster''' or '''enumeration notation''' defines a set by listing its elements between curly brackets, separated by commas:
-:{{math|1=''A'' = {{mset|4, 2, 1, 3}}}}
+:{{math|1=''A'' = {4, 2, 1, 3}}}
-:{{math|1=''B'' = {{mset|blue, white, red}}}}.
+:{{math|1=''B'' = {blue, white, red}}}.
-In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{{mset|2, 4, 6}}}} and {{math|{{mset|4, 6, 4, 2}}}} represent the same set.
+In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {{math|{2, 4, 6}}} and {{math|{4, 6, 4, 2}}} represent the same set.
 For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis '{{math|...}}'. For instance, the set of the first thousand positive integers may be specified in roster notation as
-:{{math|{{mset|1, 2, 3, ..., 1000}}}}.
+:{{math|{1, 2, 3, ..., 1000}}}.
 ====Infinite sets in roster notation====
 An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
-:{{math|{{mset|0, 1, 2, 3, 4, ...}}}},
+:{{math|{0, 1, 2, 3, 4, ...}}},
 and the set of all integers is
-:{{math|{{mset|..., −3, −2, −1, 0, 1, 2, 3, ...}}}}.
+:{{math|{..., −3, −2, −1, 0, 1, 2, 3, ...}}}.
 ===Semantic definition===

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 The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
-* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {{mset|''x'' | ''x'' is a set and ''x'' ∉ ''x''}}, cannot exist.
+* Russell's paradox shows that the "set of all sets that ''do not contain themselves''", i.e., {''x'' | ''x'' is a set and ''x'' ∉ ''x''}, cannot exist.
 * Cantor's paradox shows that "the set of all sets" cannot exist.