Set theory: basics

In short: Set theory deals with grouping objects into sets and forms one of the fundamental foundations of modern mathematics. Concepts like union, intersection, and subset appear across many mathematical fields.

In more detail: A set is a well-defined collection of distinguishable objects, called the elements of the set.

In detail

• Elements and sets: An object either belongs to a set or it doesn’t; there are no intermediate stages in classical set theory.
• Union: The union of two sets contains all elements that appear in at least one of the two sets.
• Intersection: The intersection contains only the elements that appear in both sets simultaneously.
• Subset: A set is a subset of another if all its elements are also contained in the larger set.
• Empty set: The empty set contains no elements at all and is a subset of every set.
• Venn diagrams: Circle diagrams visualize relationships between sets and make union and intersection easier to understand.
• Practical application: Set theory appears in computer science in databases and search queries, as well as in probability theory (wahrscheinlichkeitsrechnung-grundlagen, kombinatorik-grundlagen).