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...
Advanced Topics

Relations and Cartesian Products

Cartesian Product

The Cartesian product of sets \(A\) and \(B\), written as \(A \times B\), is the set of all ordered pairs \((a, b)\) where \(a \in A\) and \(b \in B\).

Relations

A relation from \(A\) to \(B\) is any subset of the Cartesian product \(A \times B\). Relations describe how elements of one set are connected to elements of another.

Types of Relations

  • Reflexive: Every element is related to itself.
  • Symmetric: If \(a\) is related to \(b\), then \(b\) is related to \(a\).
  • Transitive: If \(a\) is related to \(b\) and \(b\) to \(c\), then \(a\) is related to \(c\).

Why Are Relations Important?

Relations are used to describe connections in math, science, and even social networks!

Key Formula

\[|A \times B| = |A| \times |B|\]

Examples

  • If \(A = {1, 2}\) and \(B = {x, y}\), then \(A \times B = {(1, x), (1, y), (2, x), (2, y)}\)

  • A friendship network can be modeled as a relation between people.

In a Nutshell

Relations connect elements of sets, and the Cartesian product lists all possible pairs.

Key Terms

Cartesian Product
The set of all ordered pairs from two sets.
Relation
A rule or connection between elements of two sets.
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Functions as Special Relations