File:Relation0111.svg
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Опис
This Venn diagram is meant to represent a relation between
- two sets in set theory,
- or two statements in propositional logic respectively.
Set theory: This relation could be called subdisjoint
The relation tells, that the set is empty: =
It can be written as or as .
It tells, that all elements are within the two sets and :
Example: The set of male first names and the set of female first names are subdisjoint.
But they are not complementary sets, because some names (such as Andrea) can be given to both boys and girls.
Under this condition several set operations, not equivalent in general, produce equivalent results.
These equivalences define subdisjoint 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.
Propositional logic: The subcontrary relation
The relation tells, that the statement is never true:
It can be written as or as .
It tells, that the statements and are never false together:
Example: The statements "The president's first name could be given to a girl." and "The president's first name could be given to a boy." are subcontrary: They can not be false together. But they are not contradictory, because both statements are true, if the president's first name is e.g. Andrea.
Under this condition several logic operations, not equivalent in general, produce equivalent results.
These equivalences define subcontrary 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
∅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 |
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Датум/време | Минијатура | Димензије | Корисник | Коментар | |
---|---|---|---|---|---|
тренутна | 00:43, 8. мај 2010. | 384 × 280 (4 kB) | Watchduck | layout change | |
19:57, 26. јул 2009. | 384 × 280 (25 kB) | Watchduck | |||
18:08, 10. април 2009. | 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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