Sets

Learning Objectives

• Understand the fundamental concepts of set theory, including definitions and operations involving sets such as empty, finite, infinite sets, equal sets, and subsets.
• Calculate the union and intersection of sets, and apply properties such as commutative, associative, and distributive laws.
• Analyze the difference and complement of sets, including the application of De Morgan's laws.
• Utilize Venn diagrams to visually represent relationships between sets and operations like union, intersection, and difference.
• Examine subsets of sets, including intervals as subsets of real numbers, and understand the properties of subsets.
• Apply De Morgan's laws to relate the complement of the union and intersection of sets.
• Convert between roster form and set-builder form for set representation, and interpret mathematical conditions as sets.
• Solve practical set problems using concepts like cardinality, disjoint sets, and Venn diagrams.
• Explore the historical development of set theory, including contributions by Georg Cantor and the evolution of axiomatic set theory.

Chapter Notes

Set Theory Fundamentals

• Definition of a Set: A well-defined collection of distinct objects.
• Types of Sets:
  • Empty Set: Contains no elements, denoted by $\emptyset$ or $\{\}$.
  • Finite Set: Contains a definite number of elements.
  • Infinite Set: Contains an indefinite number of elements.
  • Equal Sets: Two sets $A$ and $B$ are equal if $A = B$.
  • Subsets: $A \subseteq B$ if every element of $A$ is also in $B$.

Union and Intersection of Sets

• Union: $A \cup B = \{x : x \in A \text{ or } x \in B\}$
  • Properties:
    • Commutative: $A \cup B = B \cup A$
    • Associative: $(A \cup B) \cup C = A \cup (B \cup C)$
    • Identity: $A \cup \emptyset = A$
• Intersection: $A \cap B = \{x : x \in A \text{ and } x \in B\}$
  • Properties:
    • Commutative: $A \cap B = B \cap A$
    • Associative: $(A \cap B) \cap C = A \cap (B \cap C)$
    • Identity: $A \cap U = A$, where $U$ is the universal set.

Difference and Complement of Sets

• Difference: $A - B = \{x : x \in A \text{ and } x \notin B\}$
• Complement: $A' = U - A$, where $U$ is the universal set.
• De Morgan's Laws:
  • $(A \cup B)' = A' \cap B'$
  • $(A \cap B)' = A' \cup B'$

Venn Diagrams

• Visual representation of set operations.
• Universal set $U$ is represented by a rectangle, and subsets by circles within.

Subsets and Intervals

• Intervals as Subsets of $\mathbb{R}$:
  • Open Interval: $(a, b) = \{x \in \mathbb{R} : a < x < b\}$
  • Closed Interval: $[a, b] = \{x \in \mathbb{R} : a \leq x \leq b\}$
  • Half-Open Intervals: $[a, b)$ or $(a, b]$

De Morgan's Laws

• Complement of Union: $(A \cup B)' = A' \cap B'$
• Complement of Intersection: $(A \cap B)' = A' \cup B'$

Historical Development of Set Theory

• Georg Cantor: Developed the modern theory of sets.
• Russell's Paradox: Highlighted contradictions in naive set theory.
• Axiomatic Set Theory: Developed by Zermelo, Fraenkel, and others to resolve paradoxes.

Set Representation: Roster and Set-Builder Form

• Roster Form: Listing elements, e.g., $\{1, 2, 3\}$.
• Set-Builder Form: Describes properties, e.g., $\{x : x \text{ is a natural number less than 4}\}$.

Universal Set and Set Complements

• Universal Set ($U$): Contains all objects under consideration.
• Complement of a Set ($A'$): Elements in $U$ not in $A$.

Cardinality and Practical Set Problems

• Cardinality: Number of elements in a set, denoted $n(A)$.
• Practical Problems: Use Venn diagrams and set operations to solve.