File:Relation1110.svg
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Contents
Summary
[edit]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: =
It can be written as or as .
It tells, that the sets and have no elements in common:
Example: The set of positive numbers and the set of negative numbers are disjoint: No number is both positive and negative.
But they are not complementary sets, because the zero is neither positive nor negative.
Under this condition several set operations, not equivalent in general, produce equivalent results.
These equivalences define disjoint sets:
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.
The relation tells, that the statement is never true:
It can be written as or as .
It tells, that the statements and are never true together:
Example: The statements "Number x is positive." and "Number x is negative." are contrary:
They can not be true together. But they are not contradictory, because both statements are false for x=0.
Under this condition several logic operations, not equivalent in general, produce equivalent results.
These equivalences define contrary statements:
Venn diagrams | written formulas |
---|---|
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
[edit] ∅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 |
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
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 22:50, 7 May 2010 | 384 × 280 (4 KB) | Watchduck (talk | contribs) | layout change | |
18:01, 26 July 2009 | 384 × 280 (9 KB) | Watchduck (talk | contribs) | |||
16:16, 10 April 2009 | 615 × 463 (4 KB) | Watchduck (talk | contribs) | {{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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File usage on Commons
The following 36 pages use this file:
- Set theory
- 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:Boolean functions as relations
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