# Set theory

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 $\scriptstyle \cup$ Bc true A ↔ A A $\scriptstyle \cup$ B A $\scriptstyle \subseteq$ Bc A$\scriptstyle \Leftrightarrow$A A $\scriptstyle \supseteq$ Bc A $\scriptstyle \cup$ Bc ¬A $\scriptstyle \or$ ¬B A → ¬B A $\scriptstyle \Delta$ B A $\scriptstyle \or$ B A ← ¬B Ac $\scriptstyle \cup$ B A $\scriptstyle \supseteq$ B A$\scriptstyle \Rightarrow$¬B A = Bc A$\scriptstyle \Leftarrow$¬B A $\scriptstyle \subseteq$ B Bc A $\scriptstyle \or$ ¬B A ← B A A $\scriptstyle \oplus$ B A ↔ ¬B Ac ¬A $\scriptstyle \or$ B A → B B B = ∅ A$\scriptstyle \Leftarrow$B A = ∅c A$\scriptstyle \Leftrightarrow$¬B A = ∅ A$\scriptstyle \Rightarrow$B B = ∅c ¬B A $\scriptstyle \cap$ Bc A (A $\scriptstyle \Delta$ B)c ¬A Ac $\scriptstyle \cap$ B B B$\scriptstyle \Leftrightarrow$false A$\scriptstyle \Leftrightarrow$true A = B A$\scriptstyle \Leftrightarrow$false B$\scriptstyle \Leftrightarrow$true A $\scriptstyle \and$ ¬B Ac $\scriptstyle \cap$ Bc A $\scriptstyle \leftrightarrow$ B A $\scriptstyle \cap$ B ¬A $\scriptstyle \and$ B A$\scriptstyle \Leftrightarrow$B ¬A $\scriptstyle \and$ ¬B ∅ A $\scriptstyle \and$ B A = Ac false A ↔ ¬A A$\scriptstyle \Leftrightarrow$¬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 seperate matrix in the box below.

## Syllogisms

Syllogisms can be described in the language of set theory.

 Barbara Celarent Darii Ferio Barbari Celaront Cesare Camestres Festino Baroco Cesaro Camestros Datisi Disamis Ferison Bocardo Felapton Darapti Calemes Dimatis Fresison Calemos Fesapo Bamalip
Venn- and Euler diagrams