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.

Test your understanding of Set Theory in this quiz focused on defining sets and their elements. Explore the basic principles that govern set definitions and the notation used. Perfect for students looking to solidify their knowledge in this area.