# Set theory

branch of mathematics that studies sets, which are collections of objects
English: Set theory is a branch of Mathematics.
It's regarded the foundation of mathematics, and closely related with logic.

## Operations on and relations between two sets

The Venn diagrams in the left matrix represent set operations - e.g. the intersection  ,
those in the right matrix represent set relations - e.g. the subset relation  , more usually represented by an Euler diagram:
The set theoretic descriptions are over the Venn diagrams:

 ∅c A = A Ac ${\displaystyle \scriptstyle \cup }$  Bc trueA ↔ A A ${\displaystyle \scriptstyle \cup }$  B A ${\displaystyle \scriptstyle \subseteq }$  Bc A${\displaystyle \scriptstyle \Leftrightarrow }$ A A ${\displaystyle \scriptstyle \supseteq }$  Bc A ${\displaystyle \scriptstyle \cup }$  Bc ¬A ${\displaystyle \scriptstyle \lor }$  ¬BA → ¬B A ${\displaystyle \scriptstyle \Delta }$  B A ${\displaystyle \scriptstyle \lor }$  BA ← ¬B Ac ${\displaystyle \scriptstyle \cup }$  B A ${\displaystyle \scriptstyle \supseteq }$  B A${\displaystyle \scriptstyle \Rightarrow }$ ¬B A = Bc A${\displaystyle \scriptstyle \Leftarrow }$ ¬B A ${\displaystyle \scriptstyle \subseteq }$  B Bc A ${\displaystyle \scriptstyle \lor }$  ¬BA ← B A A ${\displaystyle \scriptstyle \oplus }$  BA ↔ ¬B Ac ¬A ${\displaystyle \scriptstyle \lor }$  BA → B B B = ∅ A${\displaystyle \scriptstyle \Leftarrow }$ B A = ∅c A${\displaystyle \scriptstyle \Leftrightarrow }$ ¬B A = ∅ A${\displaystyle \scriptstyle \Rightarrow }$ B B = ∅c ¬B A ${\displaystyle \scriptstyle \cap }$  Bc A (A ${\displaystyle \scriptstyle \Delta }$  B)c ¬A Ac ${\displaystyle \scriptstyle \cap }$  B B B${\displaystyle \scriptstyle \Leftrightarrow }$ false A${\displaystyle \scriptstyle \Leftrightarrow }$ true A = B A${\displaystyle \scriptstyle \Leftrightarrow }$ false B${\displaystyle \scriptstyle \Leftrightarrow }$ true A ${\displaystyle \scriptstyle \land }$  ¬B Ac ${\displaystyle \scriptstyle \cap }$  Bc A ${\displaystyle \scriptstyle \leftrightarrow }$  B A ${\displaystyle \scriptstyle \cap }$  B ¬A ${\displaystyle \scriptstyle \land }$  B A${\displaystyle \scriptstyle \Leftrightarrow }$ B ¬A ${\displaystyle \scriptstyle \land }$  ¬B ∅ A ${\displaystyle \scriptstyle \land }$  B A = Ac falseA ↔ ¬A A${\displaystyle \scriptstyle \Leftrightarrow }$ ¬A
 These sets or statements have complements or negations. They are shown inside this matrix. These relations are statements, and have negations. They are shown in a separate matrix in the box below.

## Syllogisms

Syllogisms can be described in the language of set theory.

 1 Barbara Barbari Darii Ferio Celaront Celarent 2 Festino Cesaro Cesare Camestres Camestros Baroco 3 Darapti Datisi Disamis Felapton Ferison Bocardo 4 Bamalip Dimatis Fesapo Fresison Calemes Calemos

Venn- and Euler diagrams