File:Relation1011.svg
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[modifica]This Venn diagram is meant to represent a relation between
- two sets in set theory,
- or two statements in propositional logic respectively.
The relation tells, that the set is empty: =
In written formulas:
The relation tells, that the set is empty:
Under this condition, several set operations, not equivalent in general, produce equivalent results.
These equivalences define the subset relation:
Venn diagrams | written formulas |
---|---|
= | |
= | |
= | |
= | |
= | |
= | |
= | |
= |
The sign tells, that two statements about sets mean the same.
The sign = tells, that two sets contain the same elements.
Propositional logic: The logical implication
[modifica]The relation tells, that the statement is never true:
In written formulas:
The relation tells, that the statement is never true:
Under this condition, several logic operations, not equivalent in general, produce equivalent results.
These equivalences define the logical implication:
Venn diagrams | written formulas |
---|---|
Especially the last line in this table is important:
The logical implication tells, that the material implication is always true.
The material implication is the same as .
Note: Names like logical implication and material implication 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
[modifica] ∅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 |
Aquesta imatge no és apta per tenir drets d'autor perquè consta totalment d'informació que és propietat comuna i no conté cap creació original. Per tant, és en el domini públic. |
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Data/hora | Miniatura | Dimensions | Usuari/a | Comentari | |
---|---|---|---|---|---|
actual | 22:46, 7 maig 2010 | 384 × 280 (7 Ko) | Watchduck (discussió | contribucions) | layout change | |
17:59, 26 jul 2009 | 384 × 280 (12 Ko) | Watchduck (discussió | contribucions) | |||
16:13, 10 abr 2009 | 615 × 463 (4 Ko) | Watchduck (discussió | contribucions) | {{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 corresponding diagrams of th |
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Ús del fitxer
Les 37 pàgines següents utilitzen aquest fitxer:
- Set theory
- File:Logic matrix - whatsthecaserelations.svg
- 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
- Category:Black and transparent Venn diagrams (image series)
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