Set Theory

Set Theory explores the fundamental concepts of sets, including operations, relations, and applications in mathematics.

Introduction to SetsSet OperationsSubsets and Power Sets
Relations and Cartesian ProductsFunctions as Special RelationsInfinite Sets and Cardinality
Set Theory in Computer ScienceSet Theory in Everyday LifeVenn Diagrams for Problem Solving
Using Venn Diagrams on TestsBreaking Down Complex Set Problems
Introduction to SetsSet OperationsSubsets and Power Se...Relations and Cartes...Functions as Special...Infinite Sets and Ca...Set Theory in Comput...Set Theory in Everyd...Venn Diagrams for Pr...Using Venn Diagrams ...Breaking Down Comple...
Basic Concepts

Subsets and Power Sets

Subsets

A set \(A\) is a subset of set \(B\) if every element of \(A\) is also in \(B\). We write this as \(A \subseteq B\). The empty set is a subset of every set!

Proper Subsets

A proper subset is a subset that is not equal to the original set (\(A \subset B\)).

Power Set

The power set of a set is the set of all its subsets, including the empty set and the set itself. If a set has \(n\) elements, its power set has \(2^n\) elements!

Why Do Subsets Matter?

Subsets help us organize information and explore all possible combinations of items.

Examples

  • If \(A = {1, 2}\), the power set is \({\emptyset, {1}, {2}, {1,2}}\)

  • If \(B = {a, b, c}\), then \({a, b}\) is a subset of \(B\)

In a Nutshell

Subsets are parts of a set, and the power set is the set of all possible subsets.

Key Terms

Subset
A set where every element is also in another set.
Power Set
The set of all subsets of a set.
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Relations and Cartesian Products