File:Relation1011.svg
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Contents
Summary
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 subset relation
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:
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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
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 |
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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.
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Set theory: | subset | disjoint | subdisjoint | equal | complementary |
Logic: | implication | contrary | subcontrary | equivalent | contradictory |
Operations and relations in set theory and logic
∅c |
A = A |
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Ac |
true A ↔ A |
A |
A |
A |
A |
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A |
¬A A → ¬B |
A |
A A ← ¬B |
Ac |
A |
A |
A = Bc |
A |
A |
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Bc |
A A ← B |
A |
A A ↔ ¬B |
Ac |
¬A A → B |
B |
B = ∅ |
A |
A = ∅c |
A |
A = ∅ |
A |
B = ∅c | |
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¬B |
A |
A |
(A |
¬A |
Ac |
B |
B |
A |
A = B |
A |
B | |||
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A |
Ac |
A |
A |
¬A |
A |
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¬A |
∅ |
A |
A = Ac |
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false A ↔ ¬A |
A |
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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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This work is ineligible for copyright and therefore in the public domain because it consists entirely of information that is common property and contains no original authorship. |
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 22:46, 7 May 2010 | ![]() | 384 × 280 (7 KB) | Watchduck | layout change |
17:59, 26 July 2009 | ![]() | 384 × 280 (12 KB) | Watchduck | ||
16:13, 10 April 2009 | ![]() | 615 × 463 (4 KB) | Watchduck | {{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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