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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Complement (set theory) ： ウィキペディア英語版 In set theory, a complement of a set ''A'' refers to things not in (that is, things outside of) ''A''. The relative complement of ''A'' with respect to a set ''B'' is the set of elements in ''B'' but not in ''A''. When all sets under consideration are considered to be subsets of a given set ''U'', the absolute complement of ''A'' is the set of all elements in ''U'' but not in ''A''.== Relative complement ==difference (set theory), difference of two sets, relative complement, set-theoretic difference, set difference, set minus, set subtraction, set theoretic difference, setminus -->If ''A'' and ''B'' are sets, then the relative complement of ''A'' in ''B'',Halmos (1960) p.17 also termed the set-theoretic difference of ''B'' and ''A'',Devlin (1979) p.6 is the set of elements in ''B'', but not in ''A''.The relative complement of ''A'' in ''B'' is denoted according to the ISO 31-11 standard (sometimes written , but this notation is ambiguous, as in some contexts it can be interpreted as the set of all , where ''b'' is taken from ''B'' and ''a'' from ''A'').Formally: B \setminus A = \. Examples::*  ∖    =   :*  ∖    =   :* If \mathbb is the set of real numbers and \mathbb is the set of rational numbers, then \mathbb\setminus\mathbb = \mathbb is the set of irrational numbers.The following lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection.If ''A'', ''B'', and ''C'' are sets, then the following identities hold::* ''C'' ∖ (''A'' ∩ ''B'')  =  (''C'' ∖ ''A'')∪(''C'' ∖ ''B''):* ''C'' ∖ (''A'' ∪ ''B'')  =  (''C'' ∖ ''A'')∩(''C'' ∖ ''B''):* ''C'' ∖ (''B'' ∖ ''A'')  =  (''C'' ∩ ''A'')∪(''C'' ∖ ''B'')(written: ''A'' ∖ (''B'' ∖ ''C'')  =  (''A'' ∖ ''B'')∪(''A'' ∩ ''C'')  ):* (''B'' ∖ ''A'') ∩ ''C''  =  (''B'' ∩ ''C'') ∖ ''A''  =  ''B''∩(''C'' ∖ ''A''):* (''B'' ∖ ''A'') ∪ ''C''  =  (''B'' ∪ ''C'') ∖ (''A'' ∖ ''C''):* ''A'' ∖ ''A''  =  Ø:* Ø ∖ ''A''  =  Ø:* ''A'' ∖ Ø  =  ''A''==Absolute complement==Bayes' theorem and absolute set complement -->If a universe is defined, then the relative complement of in is called the absolute complement (or simply complement) of , and is denoted by or sometimes . The same set oftenBourbaki p. E II.6 is denoted by \complement_U A or \complement A if is fixed, that is:: .For example, if the universe is the set of integers, then the complement of the set of odd numbers is the set of even numbers.The following lists some important properties of absolute complements in relation to the set-theoretic operations of union and intersection.If and are subsets of a universe , then the following identities hold:: De Morgan's laws:::* \left(A \cup B \right)^=A^ \cap B^ .::* \left(A \cap B \right)^=A^ \cup B^ .: Complement laws:::* A \cup A^ =U .::* A \cap A^ =\empty .::* \empty ^ =U.::* U^ =\empty.::* \textA\subset B\textB^\subset A^.::*: (this follows from the equivalence of a conditional with its contrapositive): Involution or double complement law:::* \left(A^\right)^=A.: Relationships between relative and absolute complements:::* A \setminus B = A \cap B^c.::* (A \setminus B)^c = A^c \cup B. : Relationship with set difference:::* A^c \setminus B^c = B \setminus A. The first two complement laws above shows that if is a non-empty, proper subset of , then } is a partition of . In set theory, a complement of a set ''A'' refers to things not in (that is, things outside of) ''A''. The relative complement of ''A'' with respect to a set ''B'' is the set of elements in ''B'' but not in ''A''. When all sets under consideration are considered to be subsets of a given set ''U'', the absolute complement of ''A'' is the set of all elements in ''U'' but not in ''A''. == Relative complement == If ''A'' and ''B'' are sets, then the relative complement of ''A'' in ''B'',〔Halmos (1960) p.17〕 also termed the set-theoretic difference of ''B'' and ''A'',〔Devlin (1979) p.6〕 is the set of elements in ''B'', but not in ''A''. The relative complement of ''A'' in ''B'' is denoted according to the ISO 31-11 standard (sometimes written , but this notation is ambiguous, as in some contexts it can be interpreted as the set of all , where ''b'' is taken from ''B'' and ''a'' from ''A''). Formally : $B\; \backslash setminus\; A\; =\; \backslash .$ Examples: : *  ∖    =    : *  ∖    =    : * If $\backslash mathbb$ is the set of real numbers and $\backslash mathbb$ is the set of rational numbers, then $\backslash mathbb\backslash setminus\backslash mathbb\; =\; \backslash mathbb$ is the set of irrational numbers. The following lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection. If ''A'', ''B'', and ''C'' are sets, then the following identities hold: : * ''C'' ∖ (''A'' ∩ ''B'')  =  (''C'' ∖ ''A'')∪(''C'' ∖ ''B'') : * ''C'' ∖ (''A'' ∪ ''B'')  =  (''C'' ∖ ''A'')∩(''C'' ∖ ''B'') : * ''C'' ∖ (''B'' ∖ ''A'')  =  (''C'' ∩ ''A'')∪(''C'' ∖ ''B'') (written: ''A'' ∖ (''B'' ∖ ''C'')  =  (''A'' ∖ ''B'')∪(''A'' ∩ ''C'')  ) : * (''B'' ∖ ''A'') ∩ ''C''  =  (''B'' ∩ ''C'') ∖ ''A''  =  ''B''∩(''C'' ∖ ''A'') : * (''B'' ∖ ''A'') ∪ ''C''  =  (''B'' ∪ ''C'') ∖ (''A'' ∖ ''C'') : * ''A'' ∖ ''A''  =  Ø : * Ø ∖ ''A''  =  Ø : * ''A'' ∖ Ø  =  ''A'' ==Absolute complement== If a universe is defined, then the relative complement of in is called the absolute complement (or simply complement) of , and is denoted by or sometimes . The same set often〔Bourbaki p. E II.6〕 is denoted by $\backslash complement\_U\; A$ or $\backslash complement\; A$ if is fixed, that is: : . For example, if the universe is the set of integers, then the complement of the set of odd numbers is the set of even numbers. The following lists some important properties of absolute complements in relation to the set-theoretic operations of union and intersection. If and are subsets of a universe , then the following identities hold: : De Morgan's laws:〔 :: * $\backslash left(A\; \backslash cup\; B\; \backslash right)^=A^\; \backslash cap\; B^\; .$ :: * $\backslash left(A\; \backslash cap\; B\; \backslash right)^=A^\; \backslash cup\; B^\; .$ : Complement laws:〔 :: * $A\; \backslash cup\; A^\; =U\; .$ :: * $A\; \backslash cap\; A^\; =\backslash empty\; .$ :: * $\backslash empty\; ^\; =U.$ :: * $U^\; =\backslash empty.$ :: * $\backslash textA\backslash subset\; B\backslash textB^\backslash subset\; A^.$ :: *: (this follows from the equivalence of a conditional with its contrapositive) : Involution or double complement law: :: * $\backslash left(A^\backslash right)^=A.$ : Relationships between relative and absolute complements: :: * $A\; \backslash setminus\; B\; =\; A\; \backslash cap\; B^c.$ :: * $(A\; \backslash setminus\; B)^c\; =\; A^c\; \backslash cup\; B.$ : Relationship with set difference: :: * $A^c\; \backslash setminus\; B^c\; =\; B\; \backslash setminus\; A.$ The first two complement laws above shows that if is a non-empty, proper subset of , then } is a partition of . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「In set theory, a complement of a set ''A'' refers to things not in (that is, things outside of) ''A''. The relative complement of ''A'' with respect to a set ''B'' is the set of elements in ''B'' but not in ''A''. When all sets under consideration are considered to be subsets of a given set ''U'', the absolute complement of ''A'' is the set of all elements in ''U'' but not in ''A''.== Relative complement ==difference (set theory), difference of two sets, relative complement, set-theoretic difference, set difference, set minus, set subtraction, set theoretic difference, setminus -->If ''A'' and ''B'' are sets, then the relative complement of ''A'' in ''B'',Halmos (1960) p.17 also termed the set-theoretic difference of ''B'' and ''A'',Devlin (1979) p.6 is the set of elements in ''B'', but not in ''A''.The relative complement of ''A'' in ''B'' is denoted according to the ISO 31-11 standard (sometimes written , but this notation is ambiguous, as in some contexts it can be interpreted as the set of all , where ''b'' is taken from ''B'' and ''a'' from ''A'').Formally: B \setminus A = \. Examples::*  ∖    =   :*  ∖    =   :* If \mathbb is the set of real numbers and \mathbb is the set of rational numbers, then \mathbb\setminus\mathbb = \mathbb is the set of irrational numbers.The following lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection.If ''A'', ''B'', and ''C'' are sets, then the following identities hold::* ''C'' ∖ (''A'' ∩ ''B'')  =  (''C'' ∖ ''A'')∪(''C'' ∖ ''B''):* ''C'' ∖ (''A'' ∪ ''B'')  =  (''C'' ∖ ''A'')∩(''C'' ∖ ''B''):* ''C'' ∖ (''B'' ∖ ''A'')  =  (''C'' ∩ ''A'')∪(''C'' ∖ ''B'')(written: ''A'' ∖ (''B'' ∖ ''C'')  =  (''A'' ∖ ''B'')∪(''A'' ∩ ''C'')  ):* (''B'' ∖ ''A'') ∩ ''C''  =  (''B'' ∩ ''C'') ∖ ''A''  =  ''B''∩(''C'' ∖ ''A''):* (''B'' ∖ ''A'') ∪ ''C''  =  (''B'' ∪ ''C'') ∖ (''A'' ∖ ''C''):* ''A'' ∖ ''A''  =  Ø:* Ø ∖ ''A''  =  Ø:* ''A'' ∖ Ø  =  ''A''==Absolute complement==Bayes' theorem and absolute set complement -->If a universe is defined, then the relative complement of in is called the absolute complement (or simply complement) of , and is denoted by or sometimes . The same set oftenBourbaki p. E II.6 is denoted by \complement_U A or \complement A if is fixed, that is:: .For example, if the universe is the set of integers, then the complement of the set of odd numbers is the set of even numbers.The following lists some important properties of absolute complements in relation to the set-theoretic operations of union and intersection.If and are subsets of a universe , then the following identities hold:: De Morgan's laws:::* \left(A \cup B \right)^=A^ \cap B^ .::* \left(A \cap B \right)^=A^ \cup B^ .: Complement laws:::* A \cup A^ =U .::* A \cap A^ =\empty .::* \empty ^ =U.::* U^ =\empty.::* \textA\subset B\textB^\subset A^.::*: (this follows from the equivalence of a conditional with its contrapositive): Involution or double complement law:::* \left(A^\right)^=A.: Relationships between relative and absolute complements:::* A \setminus B = A \cap B^c.::* (A \setminus B)^c = A^c \cup B. : Relationship with set difference:::* A^c \setminus B^c = B \setminus A. The first two complement laws above shows that if is a non-empty, proper subset of , then } is a partition of .」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.