# 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
squared is equal to negative one.**i** - 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.