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

Introduction to Sets

What is a Set?

In mathematics, a set is a collection of distinct objects, called elements. Sets are often used to group numbers, shapes, or even ideas. The objects in a set can be anything—numbers, letters, or even other sets!

Notation

Sets are usually written with curly braces. For example, the set containing the numbers 1, 2, and 3 is written as \({1, 2, 3}\).

  • Element: An object in a set. If 2 is in set \(A\), we write \(2 \in A\).
  • Empty set: A set with no elements, written as \(\emptyset\) or \({}\).

Types of Sets

  • Finite set: A set with a countable number of elements, like \({a, b, c}\).
  • Infinite set: A set with endless elements, like the set of all natural numbers \({1, 2, 3, \ldots}\).

Real-World Connection

Sets are everywhere! Think about your group of friends, the toppings on a pizza, or the books on your shelf. Each of these can be thought of as a set.

Examples

  • The set of vowels in English: \({a, e, i, o, u}\)

  • The set of prime numbers less than 10: \({2, 3, 5, 7}\)

In a Nutshell

A set is a collection of distinct objects, written with curly braces.

Key Terms

Set
A collection of distinct objects.
Element
An object or member of a set.
Empty Set
A set with no elements, written as \(\emptyset\).
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Set Operations