Sets Day #1 Part I

A set is a collection of objects. Each object in the set is called an element or member of the set.

Definitions

Example. Let A be a collection of three markers.

Example. Let B be the set of students currently enrolled in this section of MAT 142.

Note: The order of listing elements in the set has no effect on the set itself. The set of the three students Bob, Ellen and Kaye is the sameas the set Ellen, Bob and Kaye.

The set of counting numbers 1, 2, 3, 4, 5, … is called the set of Natural Numbers. The set of natural numbers is denoted N.◦ Note: The three periods in this definition are called

ellipses and mean that you should continue with the established pattern.

The set containing no elements is called the empty set, or null set. The null set is denoted by { }, or by the small Greek letter phi, .f◦ Note: The curly braces in this definition are called set

braces.

Two Special Sets

I. Word Description: Let A be the set of natural numbers less than 3.

II. Set-Builder Form: ◦ A = {x | x is a natural number, x<3} ◦ In this notation the curly braces are set braces.◦ x is a variable, meaning it may take on a variety of

values.◦ The vertical bar stands for the phrase “with the

property that.”◦ A comma in this context means “and.”

III. Roster, or List Form:◦ A = { 1, 2}

Three ways of defining a set

Write set B in roster form.◦ B = { k | 2(k+1)=6 } Answer: B = { 2 }

Write set C in roster form.◦ C = { m | m is a natural number, m < 1} Answer: C = f

Write the set D in set-builder form.◦ D = {2, 3, 4, …} Answer: {x | x is a natural number, x >1}

OR {x | x is a natural number, x / 2}

Examples

A set is well-defined if any informed objective person can decide if a given element is in the set or not.

A set should be well-defined.

A is the set of goofy dogs. B is the set of GCC students whose gpa is

3.0 or greater. C is the set of good GCC students. D is the set of numbers whose square is 16.

Which of the following sets are well-defined?

Answer: B and D.

Note that D = {-4, 4}.

5 5 A means “5 is an element of the set A.” When you see the notation 5 5 A, it means

that 5 will be in the list if you write A in roster form.

More Notation

Definition of Set EqualityTwo sets are equal if they contain precisely the same elements.

{1, 2} = {2,1}

f = { } f = {0} f = {f} {0,1} = {1} 1 5 {1, 2} 0 5 {1, 2} 1 6 {1, 2} 0 6 {1, 2}

True or False True. (Same elements) True. (Same elements) False. ({0} is not empty.) False. ({f} is not empty.) False True False False True

A universal set for a particular problem is a set which contains all the elements of all the sets in the problem.

A universal set is often denoted by a capital U.

Universal Set

A = {1, 2, 3} B = {2, 4, 6, …} C = {28}

Decide on an appropriate universal set U for the following problem situation.

One answer: Let U be the set of natural numbers.

A: The set of people who are currently enrolled in a math class at GCC.

B: The set of people enrolled in a physical education class at GCC.

C: The set of people enrolled in the nursing program at GCC.

Decide on an appropriate universal set U for the following problem situation.

One answer: Let U be the set of people currently enrolled in classes at GCC.

A set is finite if it is possible, given enough time, to write down every element in the set.

A set is infinite if it is not finite.

Finite Sets versus Infinite Sets

Example. The set {1, 2, 3, …, 1000000000000000} is finite. It wouldn’t be fun to actually write every element in this set, but it is possible given enough time.

Example. The set {1, 2, 3, …} is infinite.

The cardinal number of a set A is denoted n(A).

Find the cardinal number of the following set A.◦ A = {x | x 5 N, 2 < x < 10} Answer: A = {3,4,5,6,7,8,9}, so n(A)=7.

The Cardinal Number of a finite set A is the number of elements of A.

Finite set A is equivalent to set B if n(A) = n(B).

If two sets are equivalent it means that they can be put into one-to-one correspondence. Take A={1,2,3} and B={a, b, c}. One such correspondence can be viewed graphically:

Set Equivalence for Finite Sets

1

2

3

a

b

c

True or Falsen({1,2}) = n({x,y}) True. The cardinal number of both

sets is 2.

n( ) f = n({0})False. n(f)=0 but n({0})=1. In other words, the empty set contains no elements but the set on the right contains one element, namely the number 0.

{1,2} = {x,y} False. The two sets do not contain the same elements.

{1,2} is equivalent to {x,y}. True. Both sets contain the same number of elements.

1. List the following sets in roster form.a. A={x | x is a natural number, 2x=12}b. B={k | x is a natural number, -3k=12}

2. Is the set of scary cats a well-defined set? Why or why not?

3. True or False. n({x|x is a natural number less than or equal to 5})=n({w, x, y, z}). Give a reason for your answer.

4. Give an example of two sets which are equivalent but not equal.

5. Why isn’t f equal to the number 0?

Today’s Class Quiz