Discrete Math

Study of mathematical structures that are fundamentally discrete rather than continuous.

Set TheoryLogic and ProofFunctions and Relations
Graph TheoryCombinatoricsNumber Theory
Cryptography and SecurityNetwork AnalysisScheduling and Planning
Breaking Down ProblemsUsing Visuals and Diagrams
Set TheoryLogic and ProofFunctions and Relati...Graph TheoryCombinatoricsNumber TheoryCryptography and Sec...Network AnalysisScheduling and Plann...Breaking Down Proble...Using Visuals and Di...
Basic Concepts

Set Theory

What Are Sets?

Sets are collections of distinct objects. These objects can be numbers, letters, or even other sets! Sets help us organize and classify objects by grouping them together.

Notation and Operations

Sets are typically written with curly braces, like \( S = {1, 2, 3} \). You can perform operations with sets:

  • Union (\(A \cup B\)): Combines all elements from both sets.
  • Intersection (\(A \cap B\)): Finds elements common to both sets.
  • Difference (\(A - B\)): Finds elements in one set but not the other.
  • Subset (\(A \subseteq B\)): Checks if all elements in \(A\) are also in \(B\).

Why Are Sets Important?

Sets are the foundation of most mathematical concepts in discrete math. They help us describe and analyze groups of objects in logic, computer science, and more.

Real-World Connections

Collections you use every day can be thought of as sets, like your group of friends, the books you own, or the flavors of ice cream at a shop!

Examples

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

  • The union of \({1, 2, 3}\) and \({3, 4, 5}\) is \({1, 2, 3, 4, 5}\)

In a Nutshell

Sets are collections of distinct objects, and we use them to organize and compare groups.

Key Terms

Union
Combines all elements from two sets.
Intersection
Elements common to both sets.
Subset
A set whose elements are all contained in another set.
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Logic and Proof