Sets and Set Theory

Learning Objectives for all Set Theory 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.