File:Relation1001.svg
元のファイル (SVG ファイル、384 × 280 ピクセル、ファイルサイズ: 10キロバイト)
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目次
概要[編集]
This Venn diagram is meant to represent a relation between
- two sets in set theory,
- or two statements in propositional logic respectively.
Set theory: The equivalence of sets[編集]
Two sets and are equivalent - i.e. contain the same elements - when all elements of are in , and all elements of are in .
In other words: If their symmetric difference is empty.
= | ||||||||
= |
Under this condition, several set operations, not equivalent in general, produce equivalent results.
These equivalences define equivalent sets:
= | = | = | ||||||
= | = | = |
= | = | = | ||||||
= | = | = |
= | = | = | ||||||
= | = | = |
= | = | = | ||||||
= | = | = |
The sign tells, that two statements about sets mean the same.
The sign = tells, that two sets contain the same elements.
Propositional logic: The equivalence of statements[編集]
Two statements and are equivalent - i.e. together true or together false - when implies , and implies .
In other words: If their exclusive or is never true.
Under this condition, several logic operations, not equivalent in general, produce equivalent results.
These equivalences define equivalent statements:
Especially the last line is important:
The logical equivalence tells, that the material equivalence is always true.
The material equivalence is the same as , the negated exclusive or.
Note: Names like logical equivalence and material equivalence are used in many different ways, and shouldn't be taken too serious.
The sign tells, that two statements about statements about whatever objects mean the same.
The sign tells, that two statements about whatever objects mean the same.
Set theory: | subset | disjoint | subdisjoint | equal | complementary |
Logic: | implication | contrary | subcontrary | equivalent | contradictory |
Operations and relations in set theory and logic[編集]
∅c |
A = A |
|||||||||||||
Ac Bc |
true A ↔ A |
A B |
A Bc |
AA |
A Bc |
|||||||||
A Bc |
¬A ¬B A → ¬B |
A B |
A B A ← ¬B |
Ac B |
A B |
A¬B |
A = Bc |
A¬B |
A B |
|||||
Bc |
A ¬B A ← B |
A |
A B A ↔ ¬B |
Ac |
¬A B A → B |
B |
B = ∅ |
AB |
A = ∅c |
A¬B |
A = ∅ |
AB |
B = ∅c | |
¬B |
A Bc |
A |
(A B)c |
¬A |
Ac B |
B |
Bfalse |
Atrue |
A = B |
Afalse |
Btrue | |||
A ¬B |
Ac Bc |
A B |
A B |
¬A B |
AB |
|||||||||
¬A ¬B |
∅ |
A B |
A = Ac |
|||||||||||
false A ↔ ¬A |
A¬A |
|||||||||||||
These sets (statements) have complements (negations). They are in the opposite position within this matrix. |
These relations are statements, and have negations. They are shown in a separate matrix in the box below. |
more relations | ||||
---|---|---|---|---|
|
Public domainPublic domainfalsefalse |
この著作物は、完全に常識的な情報から構成され創作性を欠くために、著作権発生の資格がなく、故にパブリックドメインの状態にあります。 |
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日付と時刻 | サムネイル | 寸法 | 利用者 | コメント | |
---|---|---|---|---|---|
現在の版 | 2010年5月7日 (金) 22:44 | 384 × 280 (10キロバイト) | Watchduck (トーク | 投稿記録) | layout change | |
2009年7月26日 (日) 17:58 | 384 × 280 (20キロバイト) | Watchduck (トーク | 投稿記録) | |||
2009年4月10日 (金) 16:10 | 615 × 463 (4キロバイト) | Watchduck (トーク | 投稿記録) | ==Description== {{Information |Description={{en|1=Venn diagrams of the sixteen 2-ary Boolean '''relations'''. Black (0) marks empty areas (compare empty set). White (1) means, that there ''could'' be something. There are correspondin |
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ファイルの使用状況
以下の 36 ページがこのファイルを使用しています:
- Set theory
- File:Equivalencia de clases.JPG
- File:Relation0000.svg
- File:Relation0001.svg
- File:Relation0010.svg
- File:Relation0011.svg
- File:Relation0100.svg
- File:Relation0101.svg
- File:Relation0110.svg
- File:Relation0111.svg
- File:Relation1000.svg
- File:Relation1001.svg
- File:Relation1010.svg
- File:Relation1011.svg
- File:Relation1100.svg
- File:Relation1101.svg
- File:Relation1110.svg
- File:Relation1111.svg
- File:Venn0000.svg
- File:Venn0001.svg
- File:Venn0010.svg
- File:Venn0011.svg
- File:Venn0100.svg
- File:Venn0101.svg
- File:Venn0110.svg
- File:Venn0111.svg
- File:Venn1000.svg
- File:Venn1001.svg
- File:Venn1010.svg
- File:Venn1011.svg
- File:Venn1100.svg
- File:Venn1101.svg
- File:Venn1110.svg
- File:Venn1111.svg
- Template:Operations and relations in set theory and logic
- Template:Operations and relations in set theory and logic; some
グローバルなファイル使用状況
以下に挙げる他のウィキがこの画像を使っています:
- ca.wikipedia.org での使用状況
- de.wikibooks.org での使用状況
- es.wikipedia.org での使用状況
- ja.wikipedia.org での使用状況