Set Theory Unit 2 Quiz

Set Theory Overview

• A set is a well-defined collection of distinct objects, denoted by uppercase letters (e.g., S, A, B) and its elements by lowercase letters (e.g., a, b, c).
• Elements are enclosed in curly brackets: S = {2, 3, 4}. The membership of an element in a set is indicated by the symbol ∈ (e.g., 2 ∈ S).
• Important characteristics of sets:
  • Existence of a rule for membership determination.
  • Order of elements is irrelevant.
  • Elements can represent quantitative or qualitative characteristics (e.g., C = {male, 64 inches}).

Methods of Representing Sets

• Tabular Form: Elements are listed within braces and separated by commas (e.g., A = {2, 4, 6, 8...}).
• Set-Builder Form: Represents large sets that cannot be listed conveniently, using properties to define elements (e.g., A = {x: x is an integer}).

Cardinality of a Set

• Cardinality indicates the number of elements in a set, denoted as n(S) or |S|.
• Example: For the set S = {2, 3, 4}, the cardinality is 3.

Types of Sets

• Null/Empty Set: Contains no elements, denoted by { } or Ø, with cardinality 0.
• Singleton Set: Contains a single element (e.g., {2}, {a}, {0}).
• Finite Set: Has a limited number of elements; the empty set Ø is finite.
• Infinite Set: Contains uncountable or infinite elements (e.g., the set of all positive integers).
• Universal Set: Contains all elements of interest in a study and all possible subsets.
• Subset: A set A is a subset of B (A ⊆ B) if all elements of A are also in B.
• Equal Sets: Two sets A and B are equal (A = B) if they contain exactly the same elements.
• Disjoint Sets: Two sets that share no common elements (e.g., A = {1, 2, 3} and B = {6, 8, 9}).
• Power Set: Collection of all subsets of a set, denoted P(A). For a set with n elements, the number of subsets is 2^n.

Properties of Sets

• The null set is a subset of all sets.
• Every set is a subset of itself.
• Union and intersection properties:
  • A ∪ (B ∪ C) = (A ∪ B) ∪ C
  • A ∩ (B ∩ C) = (A ∩ B) ∩ C
  • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
• A ∪ Ø = A

Venn Diagrams

• Venn diagrams visually represent sets and their relationships, including intersections and unions.