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15. Sets

The chapter provides a comprehensive introduction to sets, including their definitions, representations, and various operations like union, intersection, and difference. It also discusses set identities and the power set concept, along with notable properties such as equality, subsets, and cardinality of sets.

Sections

Sets

This section introduces the concept of sets, their definitions, and various operations associated with them.

15 Section Overview

Start current section content and materials

15.1 Definition of Sets

This section introduces the concept of sets as unordered collections of objects, explaining their definitions, notations, types, and fundamental operations.

15.2 Methods of Expressing a Set

This section discusses two primary methods for expressing a set: the roster method and the set-builder method.

15.3 Special Sets

This section introduces various special sets such as the empty set and singleton set, their definitions, and emphasizes their differences.

15.4 Definitions in the Context of Sets

This section explores foundational definitions related to sets, including their characteristics, equality, subsets, and the concepts of cardinality and power sets.

15.5 Cardinality of a Set

This section focuses on defining the cardinality of a set, exploring its significance in mathematics, and introducing related concepts such as power sets and types of sets.

15.6 Power Set

The section introduces the concept of power sets, explaining how they consist of all subsets of a set and the significance of cardinality in determining their size.

15.7 Set Operations

In this section, we explore fundamental operations involving sets, including union, intersection, difference, and Cartesian products, along with their theoretical foundations.

15.8 Set Identities

This section covers the definition and various identities related to set theory.

15.9 Proving Set Identities

This section explores the principles and methods for proving set identities, focusing on key definitions and operations in set theory.

Learning Objectives

  • A set is defined as an unordered collection of objects.

  • The roster method and set builder form are two methods to express a set.

  • The cardinality of a set is the count of its elements, distinguishing between finite and infinite sets.

Key Concepts

Set

An unordered collection of distinct objects.

Union

The union of two sets A and B is the set of elements that are in either A, B, or both.

Intersection

The intersection of two sets A and B consists of elements that are in both A and B.

Power Set

The power set of a set S is the set of all possible subsets of S, including the empty set and S itself.

Cardinality

The number of elements in a set, denoted by |S|.

Subset

A set A is a subset of set B if every element of A is also an element of B.

Proper Subset

A set A is a proper subset of set B if A is a subset of B and there exists at least one element in B that is not in A.

Practice Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting
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