Set theory
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English:Set theoryis a branch of Mathematics.
It's regarded the foundation of mathematics, and closely related withlogic.
It's regarded the foundation of mathematics, and closely related withlogic.
Operations onandrelations betweentwo sets
[edit]TheVenn diagramsin the left matrix representset operations- e.g. theintersection,
those in the right matrix representset relations- e.g. thesubset relation,more usually represented by anEuler diagram:
The set theoretic descriptions are over the Venn diagrams:
∅c |
A = A |
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AcBc |
true A ↔ A |
AB |
ABc |
AA |
ABc |
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ABc |
¬A¬B A → ¬B |
AB |
AB A ← ¬B |
AcB |
AB |
A¬B |
A = Bc |
A¬B |
AB |
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Bc |
A¬B A ← B |
A |
AB A ↔ ¬B |
Ac |
¬AB A → B |
B |
B =∅ |
AB |
A =∅c |
A¬B |
A =∅ |
AB |
B =∅c | |
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¬B |
ABc |
A |
(AB)c |
¬A |
AcB |
B |
Bfalse |
Atrue |
A = B |
Afalse |
Btrue | |||
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A¬B |
AcBc |
AB |
AB |
¬AB |
AB |
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¬A¬B |
∅ |
AB |
A = Ac |
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false A ↔ ¬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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Syllogisms
[edit]Syllogismscan be described in the language of set theory.
1 | ![]() Barbara |
![]() Barbari |
![]() Darii |
![]() Ferio |
![]() Celaront |
![]() Celarent |
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2 | ![]() Festino |
![]() Cesaro |
![]() Cesare |
![]() Camestres |
![]() Camestros |
![]() Baroco |
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3 | ![]() Darapti |
![]() Datisi |
![]() Disamis |
![]() Felapton |
![]() Ferison |
![]() Bocardo | ||||||||
4 | ![]() Bamalip |
![]() Dimatis |
![]() Fesapo |
![]() Fresison |
![]() Calemes |
![]() Calemos |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Syllogism-Set-Diagrams.svg/600px-Syllogism-Set-Diagrams.svg.png)
Partitions
[edit]Various files
[edit]-
Venn 0101 0101.svg
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Venn 0011 0011.svg
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Venn 0000 1111.svg
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Venn 0000 0001.svg
All 256 Venn diagrams of this kind can be foundhere