Sets and Set Theory

Learning Topics: 

Visual approach for basic definitions and notation, types of sets, equality, Venn diagrams, subsets, Universal set, set-builder notation, complement, intersection and union.

This page lists the Learning Objectives for all lessons in Unit 15.


Introduction to Sets

The student will be able to:

  • Define set, inclusive, element, object, and roster notation.
  • Identify the elements of a given set.
  • Describe conventions used to list sets.
  • List the elements of a set using roster notation.
  • List the elements of a set by describing the set and the rules that its elements follows.
  • Recognize when to describe a set and its elements instead of listing it in roster notation.
  • Apply basic set concepts to complete five interactive exercises.

Basic Notation

The student will be able to:

  • Define member.
  • Identify basic set notation which indicates whether an object is, or is not an element of a set.
  • Write set notation to indicate whether an object is, or is not, an element of a set..
  • Describe the meaning of basic set notation.
  • Determine if a given element is, or is not a member of a set.
  • Apply basic set notation to complete five interactive exercises.

Types of Sets

The student will be able to:

  • Define ellipsis, finite, infinite, countable, finite set, infinite set, empty, and null set.
  • Determine if a given set is finite or infinite.
  • List a given set using roster notation or by describing it, including an ellipsis when appropriate..
  • Classify several sets as finite or infinite.
  • Recognize the difference between a finite set and an infinite set.
  • Describe the difference between a finite set and an infinite set.
  • Determine if a set is empty (null).
  • List a set as null using proper notation.
  • Describe examples of null sets from the real world.
  • Apply types of sets to complete five interactive exercises.

Set Equality

The student will be able to:

  • Define equality and equal.
  • Describe the meaning of equal sets.
  • Determine if two or more sets are equal by examining their elements.
  • Indicate if two or more sets are equal or not not equal by writing the proper notation.
  • Recognize that the order in which elements appear in a set is not important.
  • Determine which sets are equal from a given list of sets.
  • Determine which sets are not equal from a given list of sets.
  • Apply equality concepts to complete five interactive exercises.

Venn Diagrams

The student will be able to:

  • Define Venn diagram, intersection, and union.
  • Recognize that a Venn diagram is a visual representation of a set.
  • Describe the procedure for drawing and labeling a Venn diagram to represent a set and the elements it contains.
  • Describe the procedure for drawing and labeling a Venn diagram to represent the intersection of two sets.
  • Describe the procedure for drawing and labeling a Venn diagram to represent the union of two sets.
  • Recognize the difference between the intersection and the union of two sets.
  • List the intersection of two sets using proper notation.
  • List the union of two sets using proper notation.
  • Given the roster notation of two sets, draw and label a Venn diagram to show their intersection.
  • Given the roster notation of two sets, draw and label a Venn diagram to show their union.
  • Recognize that a Venn diagram shows the relationship between two sets.
  • Apply Venn diagrams to complete five interactive exercises.

Subsets

The student will be able to:

  • Define subset, proper subset, and equivalent sets.
  • Indicate that one set is a subset of another by writing the proper notation.
  • Indicate that one set is not a subset of another by writing the proper notation.
  • Describe the procedure for drawing and labeling a Venn diagram to show the relationship between a set and its subset.
  • Identify the relationships between sets and their subsets using a Venn diagram.
  • Define equivalent sets in terms of subsets.
  • List all subsets for a given set using proper notation.
  • Recognize that the empty (null) set is a subset of all sets.
  • Recognize the difference between a subset and a proper subset.
  • Identify a proper subset of a given set.
  • Identify a subset of a given set.
  • Examine patterns in the number of subsets of a given set.
  • Describe the relationship between the number of subsets of a set and the number of elements it has.
  • List the formula for finding the number of subsets of a set with n elements.
  • Apply all subset concepts to complete five interactive exercises.

Universal Set

The student will be able to:

  • Define Universal set, overlapping, and disjoint.
  • Describe the procedure for drawing and labeling a Venn diagram to represent the Universal set.
  • List the Universal set for a given set using proper notation.
  • Recognize that every set is a subset of the Universal set.
  • Distinguish between overlapping, disjoint and subsets as they relate to a Universal set.
  • Recognize that a universal set includes everything under consideration, or everything that is relevant to the problem you have.
  • Given the roster notation of two sets, and their universal set, draw and label a Venn diagram to represent the relationship between all sets.
  • Apply the Universal set to complete five interactive exercises.

Set-Builder Notation

The student will be able to:

  • Define set-builder notation.
  • Define common types of numbers including the set of integers, whole, counting, natural, rational, real, imaginary and complex.
  • Recognize that the common types of numbers listed above are infinite sets.
  • Define i.
  • List or describe all elements in a given set written with set-builder-notation.
  • Describe the general form used for set-builder notation.
  • Read and write sets using set builder-notation
  • Recognize the  importance of indicating inclusivity when using set-builder notation.
  • Explain the meaning of a set given in set-builder-notation.
  • Classify a set given in set-builder notation as all or part of a set of numbers (integers, whole, counting, natural, rational, real, imaginary or complex numbers).
  • Explain why i squared is equal to negative one.
  • Evaluate a set of real numbers to solve equations for the unknown value.
  • Apply set-builder notation to complete five interactive exercises.

Complement

The student will be able to:

  • Define complement, universe, and A-prime.
  • Define union and intersection in terms of a complement.
  • List the notation for a set and its complement.
  • Identify the complement of a set shown in a Venn Diagram.
  • Examine the logical and visual relationship between a set and its complement using Venn diagrams.
  • Examine the logical and visual relationship between a set and its complement using set-builder notation.
  • Examine the complement of a single set, two disjoint sets, and two overlapping sets.
  • Apply set notation and set-builder notation to list sets and their complements.
  • Express the complement of a set using set notation in terms of a union.
  • Express the complement of a set using set notation in terms of an intersection.
  • Recognize that there are several different ways to represent the complement of a set.
  • Distinguish between all notation used to represent the complement of a set.
  • Given the set-builder notation of a set and its universe, draw and label a Venn diagram to represent their relationship.
  • Apply complement concepts and notation to complete five interactive exercises.

Intersection

The student will be able to:

  • Define the intersection of two sets.
  • Describe the intersection of two sets by examining a Venn diagram.
  • List the intersection of two sets using proper set notation.
  • Describe the procedure for drawing Venn diagrams to illustrate the intersection of two sets.
  • Examine the intersection of disjoint sets, overlapping sets, and subsets through Venn diagrams.
  • Express the intersection of two sets in terms of a subset.
  • Redefine the empty set in terms of intersection.
  • Redefine disjoint sets in terms of intersection.
  • Express the intersection of two sets using set-builder notation.
  • Summarize the procedure for drawing the Intersection of one set contained within another.
  • Apply intersection concepts, notation and procedures to complete five interactive exercises.

Union

The student will be able to:

  • Define the union of two sets.
  • Describe the union of two sets by examining a Venn diagram.
  • List the union of two sets using set notation.
  • Compare the union and intersection of two given sets using set notation.
  • Describe the procedure for drawing Venn diagrams to illustrate the union of two sets.
  • Examine the union of disjoint sets, overlapping sets, and subsets through Venn diagrams.
  • Express the union of two sets using set-builder notation.
  • Apply union concepts, notation and procedures to complete five interactive exercises.

Practice Exercises

The student will be able to:

  • Examine ten interactive exercises for all topics in this unit.
  • Identify the concepts, notation and procedures needed to complete each practice exercise.
  • Compute all answers and solve all problems by applying appropriate concepts, notation and procedures.
  • Self-assess knowledge and skills acquired from the instruction provided in this unit.

Challenge Exercises

The student will be able to:

  • Evaluate ten challenging, non-routine exercises for all topics in this unit.
  • Analyze each problem to identify the given information.
  • Formulate a strategy for solving each problem.
  • Apply strategies to solve problems and write answers.
  • Synthesize all information presented in this unit.
  • Develop strong problem-solving skills and the ability to handle non-routine problems.

Solutions

The student will be able to:

  • Examine the solution for each exercise presented in this unit.
  • Compare solutions to completed exercises.
  • Identify which solutions need to be reviewed.
  • Identify and evaluate incorrect answers to exercises from this unit.
  • Amend and label original answers.
  • Identify areas of strength and weakness.
  • Decide which concepts, notation, and procedures need to be reviewed from this unit.