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

Set Operations

Combining and Comparing Sets

Sets can be combined and compared using different operations. These help us see how sets overlap or differ.

Union

The union of two sets \(A\) and \(B\), written as \(A \cup B\), is the set of all elements that are in \(A\) or \(B\) or both.

Intersection

The intersection of two sets \(A\) and \(B\), written as \(A \cap B\), is the set of elements that are in both \(A\) and \(B\).

Difference

The difference of two sets \(A\) and \(B\), written as \(A - B\) or \(A \setminus B\), is the set of elements that are in \(A\) but not in \(B\).

Complement

The complement of a set \(A\) (relative to a universal set \(U\)), written as \(A'\) or \(A^c\), is the set of all elements in \(U\) not in \(A\).

Visualizing with Venn Diagrams

Venn diagrams are a fun way to show set operations with overlapping circles.

Examples

  • If \(A = {1, 2, 3}\) and \(B = {3, 4, 5}\), then \(A \cup B = {1, 2, 3, 4, 5}\)

  • If \(A = {a, b, c}\) and \(B = {b, c, d}\), then \(A \cap B = {b, c}\)

In a Nutshell

Set operations let us combine or compare sets in different ways.

Key Terms

Union
The set of all elements in either set.
Intersection
The set of elements common to both sets.
Difference
The set of elements in one set but not the other.
Complement
All elements not in the given set, relative to a universal set.
Next
Subsets and Power Sets