File:Relation0110.svg
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摘要
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 complementary relation
Two sets and are complementary,
when they are disjoint and subdisjoint (no elements are inside both and no elements are outside both of them),
so when all elements are either in set or in set .
In other words: When the complement of their symmetric difference is empty.
= | ||||||||
= |
Under this condition, several set operations, not equivalent in general, produce equivalent results.
These equivalences define complementary sets:
= | = | = | ||||||
= | = | = |
= | = | = | ||||||
= | = | = |
= | = | = | ||||||
= | = | = |
= | = | = | ||||||
= | = | = |
The sign tells, that two statements about sets mean the same.
The sign = tells, that two sets contain the same elements.
Operations and relations in set theory and logic
∅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 | ||||
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Public domainPublic domainfalsefalse |
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目前 | 2010年5月7日 (五) 22:42 | 384 × 280(4 KB) | Watchduck | layout change | |
2009年7月26日 (日) 17:57 | 384 × 280(30 KB) | Watchduck | |||
2009年4月10日 (五) 16:06 | 615 × 463(4 KB) | Watchduck | ==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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