Chapter 6 - Sets and Countingfaculty.southwest.tn.edu/hprovinc/content/Materials/Lecture Notes... · Chapter 6 - Sets and Counting ... Complement of a Set – For any set A within

1 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College

Chapter 6 - Sets and Counting

Section 6.1 Sets and Set Operations

Set Terminology and Notation

A set is a well-defined collection of objects.

Sets are usually denoted by upper case letters such as A, B, C, …

The objects of a set are called elements, or members, of a set.

Elements are usually denoted by lower case letters such as a, b, c, …

The elements of a set may be displayed by listing each element between braces.

For example, using roster notation, the set A consisting of the first three letters of the

English alphabet is written A = {a, b, c}

The set B of all letters of the alphabet may be written B = {a, b, c, …, z}

Another notation commonly used is set-builder notation.

Here, a rule is given that describes the definite property or properties an object x must satisfy to

qualify for membership in the set.

For example, the set B of all letters of the alphabet may be written as

B = {x | x is a letter of the English alphabet} and is read “B is the set of all elements of x

such that x is a letter of the English alphabet.”

There is also terminology regarding to whether an element belongs to a set or not:

If a is an element of a set A, we write a A and read “a belongs to A” or “a is an

element of A.”

If a is not an element of a set A, we write a A and read “a does not belong to A” or

“a is not an element of A.”

Designating Sets- A set is a collection of objects. The objects belonging to the set are called the

elements, or members of the set.

Word Description – The set of even numbers less than 10

Listing Method – {2,4,6,8}

Set Notation – {x | x is an even number less than 10}


Set Equality

Two sets A and B are equal, written A = B, if and only if they have exactly the same elements.

Example: Let A, B, and C be the sets

Then, A = B since they both contain exactly the same elements.

Note that the order in which the elements are displayed is irrelevant.

Also, A ≠ C since u A but u C.

Similarly, we conclude that B ≠ C.

Subset

If every element of a set A is also an element of a set B, then we say that A is a subset of B and

write A B.

Set Equality – Suppose A and B are sets. Then are both true.

Example: Let U, M, and N be the sets

Subsets of Sets – Set A is a subset of set B is every element of A is also an element of B.

For example in the above example set M is a subset of U, the notation for this is and set N is a

subset of U, . Furthermore when something is not a subset like N is not a subset of M, the

notation is .

Number of Subsets – The number of subsets of a set with n elements is

Where n is the number of elements

Sets are commonly given names or symbols:

E is the set of all letters of the English alphabet

is the set containing no elements is called the empty set or null set.

The empty set, Ø, is a subset of every set

∉ this means is not an element of

∉

∉

∈ this means is an element of

∈

∈

{ , , , , }

{ , , , , }

{ , , , }

A a e i o u

B a i o e u

C a e i o

{ , , , , }

{ , , , , }

{ , , , }

A a e i o u

B a i o e u

C a e i o


Example – Determine whether one set is a subset of another

Write or in each blank

a) {3,4,5,6} _____{3,4,5,6,8}

b) {1,2,6} _____ {2,4,6,8}

c) {5,6,7,8} ______ {5,6,7,8}

Example – Listing all subsets of a Set

Find all possible subsets of each set

a) {7,8}

b) {a,b,c}

Example - Example – Give a complete listing of all the elements of each of the following sets

a) The set of counting numbers between six and thirteen

b)

c) {x | x is a counting number between 6 and 7}

Example – Fill in the blank with ∈ or ∉

a) 5 ____ {2,4,5,7}

b) -12 ____ {3,8,12,18}

c) 3 ____ {2,3,4,6}

d) m ____ {l,m,n,o,p}


Set of Numbers

Natural or Counting Numbers {1,2,3,4,…}

Whole Numbers {0,1,2,3,4,…}

Integers {…,-3,-2,-1,0,1,2,3,…}

Rational Numbers {

|

Real Numbers |

Irrational Numbers Examples √ √

|

Example –

1. Are A and B equal sets? A = {-4,3,2,5} and B = {-4,0,3,2,5}

2. Are the following true or false

a. {3} = {x | x is a counting number between 1 and 5}

b. {x | x is a negative natural number} = {y | y is a number that is both rational and

irrational}

Universal Set

A universal set is the set of all elements of interest in a particular discussion.

It is the largest set in the sense that all sets considered in the discussion of the problem are

subsets of the universal set.

For Example:

If the problem at hand is to determine the ratio of female to male students in a college, then the

logical choice of a universal set is the set consisting of the whole student body of the college

If the problem is to determine the ratio of female to male students in the business department

of the college, then the set of all students in the business department may be chosen as the

universal set


Intersection of Sets – The intersection of sets A and B, written A⋂B, is the set of elements common in

both A and B, or | ∈ ∈

Union of Sets – The union of sets A and B, written A⋃B, is the set of elements belonging to wither of the

sets, or | ∈ ∈

Disjoint Sets – Two sets have no elements in common, sets A and B are disjoint if A⋂B = ø.

A B

Intersection of Sets – “AND”

The Venn diagram to the left shows two sets A and B the

intersection of the two sets is the gray portion, , U is

the universe

A B

Union of Sets – “OR”

The Venn diagram to the left shows two sets A and B the

union of the two sets is the gray portion, , U is the

universe

A B

U

U

U

Disjoint Sets – “NONE”

The Venn diagram to the left shows two sets A and B that

are disjoint having no elements in common, U is the

universe


Example – Find each intersection

a)

b)

c)


Example – Find each union

a)

b)

c)


Venn diagrams or set diagrams are diagrams that show all hypothetically possible logical relations

between a finite collection of sets (groups of things).

Example -

Complement of a Set – For any set A within the universal set U, the complement of A, written A’, is the

set of elements of U that are not elements of A, That is, | ∈ ∉

U

B

A

7 9

10

8

2 3

6

1 4

5

A A’

7 9

10

U

A

B

http://en.wikipedia.org/wiki/File:Venn-diagram-AB.svg


Set Complementation

If U is a universal set and A is a subset of U, then

a. Uc = Ø b. Øc = U

c. (Ac)c = A d. A Ac = U

e. A Ac = Ø

Example – Finding Complements

U’ =

M’=

N’=

Example – Finding the Intersection and Union of Complements and draw a Venn Diagram and shade the

portion that is being asked for.

a)

b)


c)

d)

Properties of Set Operations

Let U is a universal set. If A , B, and C are arbitrary subsets of U, then we have the following laws:

Commutative law for union

A B = B A

Commutative law for intersection

A B = B A

Associative law for union

A (B C) = (A B) C

Associative law for intersection

A (B C) = (A B) C

Distributive law for union

A (B C) = (A B) (A C)

Distributive law for intersection

A (B C) = (A B) (A C)

De Morgan’s Laws

(A B)c = Ac Bc

De Morgan’s Laws

(A B)c = Ac Bc


Example – Describe using math language what is shown for each of the following

a)

b)


Example - Applied Example: Automobile Options

Let U denote the set of all cars in a dealer’s lot, and let

A = {x U | x is equipped with automatic transmission}

B = {x U | x is equipped with air conditioning}

C = {x U | x is equipped with side air bags}

Find an expression in terms of A, B, and C for each of the following sets:

The set of cars with at least one of the given options.

The set of cars with exactly one of the given options.

The set of cars with automatic transmission and side air bags but no air conditioning.

Example – Place the elements in the proper location in a Venn Diagram

and shade the portion corresponding to (A’⋂B’)⋂C


Section 6.2 The Number of Elements in a Finite Set

Counting Elements of a Set

The number of elements in a finite set is determined by simply counting the elements in the set.

If A is a set, then n(A) denotes the number of elements in A.

For example, if A = {1, 2, 3, …,20} B = {a, b} C = {8}

then n(A) = 20, n(B) = 2, and n(C) = 1.

The empty set has no elements in it, so n(Ø) = 0.

If A and B are disjoint sets, then n(A B) = n(A) + n(B)

In the general case, A and B need not be disjoint, which leads us to the formula

n(A B) = n(A) + n(B) – n(A B)

Example - If A = {a, c, d} and B = {b, e, f, g}, find the following

a) A B

b) n(A) + n(B)

c) A B

Counting Elements of a Set

Similar rules can be derived for cases involving more than two sets.

For example, if we have three sets A, B, and C, we find that

n(A B C) = n(A) + n(B) + n(C) – n(A B)

– n(A C) – n(B C) + n(A B C)


Example - Applied Example: Consumer Surveys

In a survey of 100 coffee drinkers, it was found that 70 take sugar, 60 take cream, and 50 take both

sugar and cream with their coffee. How many coffee drinkers take sugar, cream, or both with their

coffee?


Section 6.3 The Multiplication Principle

The Fundamental Principle of Counting

The Multiplication Principle

Suppose there are m ways of performing a task T1 and n ways of performing a task T2.

Then, there are mn ways of performing the task T1 followed by the task T2.

Example: Three roads connect town A and town B, and two roads connect town B and town C. Use the

multiplication principle to find the number of ways a journey from town A to town C via town B may be

completed.

Generalized Multiplication Principle

Suppose a task T1 can be performed in N1 ways, a task T2 can be performed in N2 ways, …, and,

finally, a task Tn can be performed in Nn ways.

Then, the number of ways of performing tasks T1, T2, …, Tn in succession is given by the product

N1 N2 ··· Nn

Example: A coin is tossed 3 times, and the sequence of heads and tails is recorded. Use the generalized

multiplication principle to determine the number of possible outcomes of this activity


Example: A combination lock is unlocked by dialing a sequence of numbers: First to the left, then to the

right, and to the left again. If there are ten digits on the dial, determine the number of possible

combinations.

Example:– Counting License Plates

In some states, auto license plates have contained three letters followed by three digits. How many such

licenses are possible?

Example: Building Numbers from a Set of Digits

A four-digit number is to be constructed using only digits from the set {1, 2, 3}

a) How many such numbers are possible?

b) How many of these numbers are odd and less than 2000?


Example: Counting Three Digit Numbers with Restrictions

How many non-repeating odd three-digit counting numbers are there?

Example: Electing Club officers with Restrictions

In how many ways can Club N elect a president and a secretary if no one may hold more than one office

and the secretary must be a man? Consider a club N with five members: N = {Alan, Bill, Cathy, David,

Evelyn} or, in abbreviated form, N = {A, B, C, D, E}


Section 6.4 Permutations and Combinations

Factorial Formula – For any counting number n, the quantity n factorial is given by

The zero factorial is 1.

Example – Evaluating Expressions Containing Factorials

Evaluate each expression.

a) 3!

b) 6!

c) (6-3)!

d) 6!-3!

e)

f) (

)

g) 15!

h) 100!

Arrangements of n Distinct objects n!

The total number of different ways to arrange n distinct objects is n!

Example: Arranging Essays

Erika Berg has seven essays to include in her English 1A folder. In how many different orders can she

arrange them?

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Example: Arranging Preschoolers

Lynn Damme is taking thirteen preschoolers to the park. How many ways can the children line up, in

single file, to board the van?

Arrangements of n Objects Containing Look-Alikes

The number of distinguishable arrangements of n objects, where one or more subsets consist of look-

alikes (say n1 are of one kind n2 are of another kind… and nk are of yet another kind), is given by

Example – Determine the number of distinguishable arrangements of the letters in each word.

a) HEEDLESS

b) NOMINEE

Permutations – arrangements are often called permutations, the number of permutations of n distinct

things taken r at a time is denoted nPr Since the number of objects being arranged cannot exceed the

total number available, we assume for our purposes here that r ≤ n. Applying the fundamental counting

principle to arrangements of this type gives:

nPr =

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Permutations are to evaluate the number of arrangements of n things taken r at a time, where

repetitions are not allowed, and the order of the items is important.

Factorial Formula for Permutations

The number of permutations, or arrangements, of n distinct things taken r at a time, where r ≤ n, can be

calculated as:

nPr

Alternative Notations are P(n,r) and

For Example 4P2 means “the number of permutations of 4 distinct things taken 2 at a time”.

Example: Using the Factorial Formula for Permutations

Evaluate each permutation

a) 4P2

b) 8P5

c) 5P5

Notice that 5P5 is equal to 5! It is true for all whole numbers n that nPn = n!

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Example: Calculating Permutations Directly

Evaluate each permutation

Using a graphing calculator we can perform this calculation directly as follows for a TI-83:

For 10P6 enter in 10 then hit MATH – Scroll over to PRB and scroll down to 2 (nPr) hit enter

then enter in 6 then hit enter.

10P6 = 151200

a) 25P0

b) 18P12

Example:– Building Numbers from a Set of Digits

How many non-repeating three-digit numbers can be written using digits from the set {3, 4, 5, 6, 7, 8}?

Note – there are 6 digits and we want to choose 3 digits.

Example: Find the number of ways a chairman, a vice-chairman, a secretary, and a treasurer can be

chosen from a committee of eight members. No member can hold more than one office.

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Example: Weaver and Kline, a stock brokerage firm, has received nine inquiries regarding new accounts.

In how many ways can these inquiries be directed to three of the firm’s account executives if each

account executive is to handle three inquiries?

Combinations – Recall that club N = {Alan, Bill, Cathy, David, Evelyn} could elect three officers in

5P3 = 60 different ways. With three member committees, on the other hand, order is not important.

The committees B, D, E and E, B, D are not different. The possible number of committees is not the

number of arrangements of size 3. Rather, it is the number of subsets of size 3 (since the order of

elements in a set makes no different)

Subsets in the new context are called combinations. The number of combinations of n things taken r at

a time (that is the number of size r subsets, given a set of size n) is written nCr Since there are n things

available and we are choosing r of them, we can read nCr as “n choose r”. The formula for evaluating

numbers of combinations

nCr

Permutations are to evaluate the number of arrangements of n things taken r at a time, where

repetitions are not allowed, and the order of the items is important.

Combinations are the number of combinations of n things taken r at a time (that is the number of size r

subsets, given a set of size n), where repetitions are not allowed, and the order is not important.

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Factorial Formula for Combinations

The number of combinations, or subsets, of n, distinct things taken r at a time, where r ≤ n, can be

calculated as

nCr

Alternative Notations are C(n,r) and and (

)

Example: Using the Factorial Formula for Combinations

Evaluate each combination

a) 9C7

b) 24C18

Example: Calculating Combinations Directly

Evaluate each combination

Using a graphing calculator we can perform this calculation directly as follows for a TI-83:

For 14C6 enter in 14 then hit MATH – Scroll over to PRB and scroll down to 2 (nCr) hit enter

then enter in 6 then hit enter.

14C6 = 3003

a) 21C15

Example: Finding the Number of Subsets

Find the number of different subsets of size 2 in the set {a, b, c, d}. List them to check the answer.

Note – there are 4 elements and we want to choose 2 elements.

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Example: Finding the Number of Possible Poker Hands

A common form of poker involves hands (sets) of five cards each, dealt from a standard deck consisting

of 52 different cards. How many different 5-card hands are possible?

Example: A Senate investigation subcommittee of four members is to be selected from a Senate

committee of ten members. Determine the number of ways this can be done.

Guidelines on Which Method to Use.

Permutations Combinations

Number of ways of selecting r items out of n items

Repetitions are NOT allowed

Order is important Order is NOT important

Arrangements of n items taken r at a time

Subsets of n items taken r at a time

nPr

nCr

Clue words: arrangement, schedule, order

Clue words: set, group, sample, selection

If selected items can be repeated, use the fundamental counting principle

If selected items cannot be repeated, and order is important, use permutations

If selected items cannot be repeated, and order is not important, use combinations.

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Example: Suppose an investor has decided to purchase shares in the stocks of two aerospace

companies, two energy development companies, and two electronics companies. In how many ways can

the investor select the group of six companies for the investment from a recommended list of five

aerospace companies, three energy development companies, and four electronics companies?

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Chapter 6 - Sets and Countingfaculty.southwest.tn.edu/hprovinc/content/Materials/Lecture Notes... · Chapter 6 - Sets and Counting ... Complement of a Set – For any set A within