Part 1

Tell students, “Today we are going to discuss sets, elements, and functions. A set is a group of elements. An element is simply a component found within a set. A function can be used to describe a set and point to the elements in the set. We will talk more about functions in a bit.” [IS.5 - Struggling Learners]

Show the following box on the board or an overhead projector:

“Now, let’s think about our real number system. Does anyone know the two disjointed number sets in our real number system?” [IS.6 - All Students] (Allow all responses, guiding the answer to rational and irrational numbers. In summarizing responses, note that rationals can be expressed as numerator divided by denominator, while irrationals cannot.)

“We have the set of rational numbers and the set of irrational numbers. Within the rational number set, we have particular embedded number sets. One set embedded in another set is called a subset. In this case, since the sets are not equal, we have what is called a proper subset.”

• “Which numbers does the set of rational numbers contain?” (integers, whole numbers, counting numbers)
• “Which sets of numbers are subsets of the rational numbers?” (integers, whole numbers, natural numbers)

“The rational numbers include all of the typical numbers we think about, including natural numbers (counting numbers), whole numbers, and integers.”

“Notice that the rational numbers also include other numbers, such as those that terminate or repeat. All rational numbers can be written in the form .”

“We can also write each of these numbers as sets. Shown below is the roster method, whereby we simply list elements of each set.”

Roster Method

Ask, “Can we illustrate the sets in another way?” (Yes; we often express some of these mathematical ideas in a special type of notation.) “Yes, first let’s draw a Venn Diagram to represent these three subsets of the real number system.”

“Next, let’s write the number sets, using something called set builder notation. [IS.7 - Struggling Learners] With set builder notation, you describe the set with symbols, using a formal notation. For example, if we were to indicate the sets of natural numbers, whole numbers, and integers, respectively, using set builder notation, we would write:

A =   the set of all x, such that x is an element of the Natural Numbers.

B =   the set of all x, such that x is an element of the Whole Numbers.

C =   the set of all x, such that x is an element of the Integers. Note that Z is used, rather than I so it is distinct from the I notation for the irrational numbers.”

“Notice that we denote each set with a letter. The letters A, B, and C serve to name the sets. By doing so, we can refer to them as Set A, Set B, and Set C. Thus far, we have seen sets with and without names. An example of a set without a name is {3, 4, 5,…}. We know this is a set. It just does not have a name. Normally, we do wish to name the set, especially when we are looking at and comparing more than one set. For example, here are three sets:

{3, 4, 5,… }

{1, 2,… }

”

“We must identify which set we are speaking about. Later on, when we look at relations, and specifically, functions of sets, it’s important to identify the set.”

“Also, notice how each notation for the number sets is put into words. We could write each set simply using words.

A = {the set of natural numbers greater than two}

B = {the set of whole numbers}

C = {the suits of a deck of cards}”

“We can go even further. If we wanted to show that one particular number is an element of the set of integers, we could write .”

“What if we wanted to identify only a part of the set of real numbers? Remember, the real numbers include both rational and irrational numbers.”

“What if we are only concerned with those real numbers greater than or equal to 4? We can represent this set, using the roster method, set-builder notation, and interval notation, as well as with several other representations.”

Summary of Notations for Real Numbers Greater Than or Equal to 4 [IS.8 - Struggling Learners]

Notation	Representation of Set	Any Specific Notes?
Set-Builder Notation	A =  (read, “the set of all real numbers, x, such that x is greater than or equal to 4”)	Since we didn’t specify that x was an element of a specific subset, we realize we are working with the set of real numbers.
Interval Notation		Since we are including the number 4, we use brackets. A number that is not included is indicated with a parenthesis. Important: Infinity is never closed with a bracket in order to show the difference between to two limits of the interval.
Graphic
Words	The set of all real numbers, greater than or equal to 4.
Other?

Interval Notation

Tell students, “Interval notation is a new one that we haven’t explored yet. Interval notation uses the endpoints of the set to describe the elements. Again, we can talk about specific elements by saying . We are saying that 7 ‘is an element of’ the set A.”

Activity 1

“It is time to practice using multiple representations of various sets. Consider these sets:

1)      The four most recent presidents of the United States

2)      Integers less than −2 and greater than −9

3)      The four seasons of the year

4)      All polygons with four or fewer sides

5)      Rational numbers greater than ”

(Answers: 1. Barack Obama, George W. Bush, Bill Clinton, George H. W. Bush;

2. –8, –7, –6, –5, –4, –3; 3. Winter, Spring, Summer, Autumn; 4. Quadrilateral, triangle; 5. {n|n > })

“The information you have just learned is useful in illustrating sets generally for a variety of uses. You might have noticed key differences in some of the sets. We have what are called finite sets and infinite sets. A finite set has a definitive number of elements listed. For example, B = {1, 2, 3} is an example of a finite set. An infinite set is a set that is not finite. The set can either be a countable infinite set, as with the set of all natural numbers, or an uncountable infinite set, as with the set of all real numbers. D = {1, 2, 3,… } and F = {all real numbers} are examples of infinite sets.”

“The focus of this lesson is on different types of number notations. Thus, it is important that you recognize and have facility working with all different representations of sets. We will now look at some of the common conventions of sets, using an example set. Suppose we are interested in the set of all whole numbers less than 8. We can represent this set in the following ways.”

Notations for the Set of All Whole Numbers Less Than 8

Name of Representation	What It Looks Like
Words	The set of all whole numbers less than 8.
Words with set name	Let A be the set of all whole numbers less than 8.
Words with set name and brackets	A = {The set of all whole numbers less than 8}.
Graphic
Number line
Roster notation without name	{0, 1, 2, 3, 4, 5, 6, 7}
Roster notation with name	A = {0, 1, 2, 3, 4, 5, 6, 7}
Set-builder notation without name
Set-builder notation with name
Interval notation	not suitable

Activity 2

Tell the class, “Choose one numeric and one non-numeric set to represent. For each set, include all of the representative forms we have discussed, as well as any others you can think of.” Allow students time to work on each set in pairs or as a small group. Encourage as many different ideas from students in the discussion and ask them to evaluate each other’s suggestions. Then ask:

• “So, what is the most commonly/widely accepted set notation, i.e., most prevalent notation?”
• “What is the most common convention used to talk about a set?”
• “We’ve looked at several different versions already, but is there one or more that are ‘just better’ than the others?”
• “What is our standard notation?”
• “Are the others just loose forms of the formal convention? Are they accepted?”
• “If we were to place a few agreed-upon set identification conventions in a file for future students to view, what would they be?”

Tell students that the four most widely-accepted notations of sets are: [IS.9 - Struggling Learners]

• Words with set name and brackets (A = {The set of all whole numbers less than 8})
• Roster notation with name (A = {0, 1, 2, 3, 4, 5, 6, 7})
• Set-builder notation with name ()
• Interval notation ([0,8))

“Notice that the most widely accepted notations are formal and include the name of the set. Now, let’s look at specific types of sets and relationships between sets. There are two types of sets we have not discussed, at least not directly. These sets are the empty set and the universal set. The empty set, denoted with either empty brackets,
{  }, or Ø, is simply a set without any elements. This might seem to be a contradiction. An example should clarify.”

Ask students to offer examples of empty sets. Lead the discussion to evaluate their examples. “Now, what if we discuss a universal set? We actually already have. The real number system is an example of a universal set. A universal set is the set that contains all other sets, including itself. Since the real number system includes the sets of rational numbers and set of irrational numbers, as well as all other real numbers, it is indeed the universal set.”

“A universal set can relate to any encompassing set, however. Let’s look at the Venn Diagram below.” Note: A Venn diagram is a graphic organizer that shows the relationship between the sets.

Explain that a Venn diagram is a very general way of representing a universal set and its subsets.

Activity 3

Tell students, “Using any sets that you wish, draw a Venn diagram to represent the universal set and any subsets.”

“Many times, we want to show the relationships between sets and/or to compare sets. We now need to talk about ways to represent relationships between sets. First, let’s think of what these representations could be. How could you relate two sets?”

“Well, we could look at the combination of the elements of both sets. We could look at the common elements. We could look at the elements contained in one, but not contained in the other. How could we display these relationships?”

“We could actually use any of the methods we’ve previously looked at.” Put the following statement on the board or an overhead projector:

         Let U be the set of integers from −4 to 8.

         Let A = {−1, 0, 3, 4} and B = {2, 3, 6, −3}

“Let’s use Venn diagrams to represent the relationships we’re looking at, discover the meaning of the symbolism, and represent the solutions in a variety of ways. Consider the table below.” [IS.10 - Struggling Learners] Distribute copies of the Set Relationships handout for student reference (M-A1-2-1_Set Relationships.docx).

Relationship	Description/Meaning	Venn Diagram	Other Representations
	This is read, “A union B,” which means “the elements found in Set A OR Set B.”
	This is read, “A intersect B,” which means “the elements common to Sets A and B,” or “the elements found in Set A AND Set B.”
	This is read, “A complement,” which means “those elements not found in Set A.” We are looking at the complement of A relative to B, not to the universal set.
	This is read, “B complement,” which means “those elements not found in Set B.” We are looking at the complement of B relative to A, not to the universal set.
	This is read, “A minus B,” which means “the elements found in set A minus the elements found in Set B”; also known as the complement of B relative to A.
	This is read, “B minus A,” which means “the elements found in set B minus the elements found in Set A”; also known as the complement of A relative to B.

Note: Complement has the same idea as “negation.” For example p and ~p represent “p” and “not p” respectively.

Activity 4

Divide students into groups of three or four. Say to the groups: “Create a universal set and at least two subsets. Choose the most effective representation of each relationship shown above. Provide a justification for why that representation makes the most sense and should be used in this case.”

“Create a short PowerPoint presentation, [IS.11 - Struggling Learners] describing when various representations for union, intersection, complement, and difference are desirable. What seems to be the most commonly accepted convention for indicating set relationships?”

Part 2

“Now that you have a firm understanding of the idea of sets, let’s use a function to create a set.”

“Before we discuss functions, we need to discuss the idea of relations. All functions are relations, but not all relations are functions. This fact is very important to note.”

“Does anyone know what a relation is? Apart from mathematics, what is a relation?” (Students may provide examples of husband, wife; child, mother; friends; grandparents, etc.)

“Now, let’s think about some relations. In particular, mapping. A mapping relationship is one where every element in the first set matches to one or more elements in the second set.” Distribute copies of the Set Examples handout (M-A1-2-1_Set Examples.docx), which contains these examples:

			Example: Pig is mapped to Sheep. These two elements are paired. Similarly, cat is not mapped.