File:Relation1001.svg
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Sammanfattning
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 equivalence of sets
Two sets and
are equivalent - i.e. contain the same elements - when all elements of
are in
, and all elements of
are in
.
In other words: If their symmetric difference is empty.
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= | ![]() | |
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= |
Under this condition, several set operations, not equivalent in general, produce equivalent results.
These equivalences define equivalent sets:
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= | ![]() |
= | ![]() |
= | ![]() |
= | = | = |
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= | ![]() |
= | ![]() |
= | ![]() |
= | = | = |
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= | ![]() |
= | ![]() |
= | ![]() |
= | = | = |
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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 equivalence of statements
Two statements and
are equivalent - i.e. together true or together false - when
implies
, and
implies
.
In other words: If their exclusive or is never true.
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Under this condition, several logic operations, not equivalent in general, produce equivalent results.
These equivalences define equivalent statements:
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Especially the last line is important:
The logical equivalence tells, that the material equivalence
is always true.
The material equivalence is the same as
, the negated exclusive or.
Note: Names like logical equivalence and material equivalence 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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Upphovsrätt kan inte tillämpas på detta verk som därmed hamnar i public domain; detta på grund av att verket enbart består av information som är allmän egendom och saknar verkshöjd. |
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Datum/Tid | Miniatyrbild | Dimensioner | Användare | Kommentar | |
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nuvarande | 8 maj 2010 kl. 00.44 | ![]() | 384 × 280 (10 kbyte) | Watchduck | layout change |
26 juli 2009 kl. 19.58 | ![]() | 384 × 280 (20 kbyte) | Watchduck | ||
10 april 2009 kl. 18.10 | ![]() | 615 × 463 (4 kbyte) | 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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