chapter 1 section 1 - slide 1 copyright © 2009 pearson education, inc. and

Chapter 1 Section 1 - Slide 1Copyright © 2009 Pearson Education, Inc.

AND

Chapter 1 Section 1 - Slide 2Copyright © 2009 Pearson Education, Inc.

Chapter 1.1Chapter 2.1, 2.2

Critical Thinking Skills

Set Concepts and Subsets

Chapter 1 Section 1 - Slide 3Copyright © 2009 Pearson Education, Inc.

WHAT YOU WILL LEARN

• Inductive and deductive reasoning processes

• Methods to indicate sets, equal sets, and equivalent sets

• Subsets and proper subsets

Chapter 1 Section 1 - Slide 4Copyright © 2009 Pearson Education, Inc.

Section 1.1

Inductive Reasoning

Chapter 1 Section 1 - Slide 5Copyright © 2009 Pearson Education, Inc.

Natural Numbers

The set of natural numbers is also called the set of counting numbers.

The three dots, called an ellipsis, mean that 8 is not the last number but that the numbers continue in the same manner.

={1,2,3,4,5,6,7,8,...}N

Chapter 1 Section 1 - Slide 6Copyright © 2009 Pearson Education, Inc.

Divisibility

If a b has a remainder of zero, then a is divisible by b.

The even counting numbers are divisible by 2. They are 2, 4, 6, 8,… .

The odd counting numbers are not divisible by 2. They are 1, 3, 5, 7,… .

Chapter 1 Section 1 - Slide 7Copyright © 2009 Pearson Education, Inc.

Inductive Reasoning

The process of reasoning to a general conclusion through observations of specific cases.

Also called induction. Often used by mathematicians and scientists to

predict answers to complicated problems.

Chapter 1 Section 1 - Slide 8Copyright © 2009 Pearson Education, Inc.

Scientific Method

Inductive reasoning is a part of the scientific method.

When we make a prediction based on specific observations, it is called a conjecture.

Chapter 1 Section 1 - Slide 9Copyright © 2009 Pearson Education, Inc.

Counterexample

In testing a conjecture, if a special case is found that satisfies the conditions of the conjecture but produces a different result, that case is called a counterexample. Only one exception is necessary to prove a

conjecture false. If a counterexample cannot be found, the

conjecture is neither proven nor disproven.

Chapter 1 Section 1 - Slide 10Copyright © 2009 Pearson Education, Inc.

Deductive Reasoning

A second type of reasoning process. Also called deduction. Deductive reasoning is the process of reasoning

to a specific conclusion from a general statement.

Chapter 1 Section 1 - Slide 11Copyright © 2009 Pearson Education, Inc.

Example: Inductive Reasoning

Use inductive reasoning to predict the next three numbers in the pattern (or sequence).

7, 11, 15, 19, 23, 27, 31,… Solution: We can see that four is added to each term to

get the following term. 31 + 4 = 35, 35 + 4 = 39, 39 + 4 = 43 Therefore, the next three numbers in the

sequence are 35, 39, and 43.

Chapter 1 Section 1 - Slide 12Copyright © 2009 Pearson Education, Inc.

Section 2.1

Set Concepts

Chapter 1 Section 1 - Slide 13Copyright © 2009 Pearson Education, Inc.

Set

A collection of objects, which are called elements or members of the set.

Listing the elements of a set inside a pair of braces, { }, is called roster form.

The symbol , read “is an element of,” is used to indicate membership in a set.

The symbol means “is not an element of.”

Chapter 1 Section 1 - Slide 14Copyright © 2009 Pearson Education, Inc.

Well-defined Set

A set which has no question about what elements should be included.

Its elements can be clearly determined. No opinion is associated its the members.

Chapter 1 Section 1 - Slide 15Copyright © 2009 Pearson Education, Inc.

Roster Form

This is the form of the set where the elements are all listed, separated by commas.

Example: Set A is the set of all natural numbers less than or equal to 25.

Solution: A = {1, 2, 3, 4, 5,…, 25} The 25 after the ellipsis indicates that the elements continue up to and including the number 25.

Chapter 1 Section 1 - Slide 16Copyright © 2009 Pearson Education, Inc.

Set-Builder (or Set-Generator) Notation

A formal statement that describes the members of a set is written between the braces.

A variable may represent any one of the members of the set.

Example: Write set B = {2, 4, 6, 8, 10} in set-builder notation.

Solution:

B x x N and x is an even number 10 .

The set of all x such that x is a natural number and x is an even number 10.£

Chapter 1 Section 1 - Slide 17Copyright © 2009 Pearson Education, Inc.

Finite Set

A set that contains no elements or the number of elements in the set is a natural number.

Example:

Set S = {2, 3, 4, 5, 6, 7} is a finite set because the number of elements in the set is 6, and 6 is a natural number.

Chapter 1 Section 1 - Slide 18Copyright © 2009 Pearson Education, Inc.

Infinite Set

An infinite set is a set where the number of elements is not or a natural number; that is, you cannot count the number of elements.

The set of natural numbers is an example of an infinite set because it continues to increase forever without stopping, making it impossible to count its members.

Chapter 1 Section 1 - Slide 19Copyright © 2009 Pearson Education, Inc.

Equal sets have the exact same elements in them, regardless of their order.

Symbol: A = B

Example: { 1, 5, 7 } = { 5, 7, 1 }

Equal Sets

Chapter 1 Section 1 - Slide 20Copyright © 2009 Pearson Education, Inc.

Cardinal Number

The number of elements in set A is its cardinal number.

Symbol: n(A)

Example: A = { 1, 5, 7, 10 }

n(A) = 4

Chapter 1 Section 1 - Slide 21Copyright © 2009 Pearson Education, Inc.

Equivalent Sets

Equivalent sets have the same number of elements in them.

Symbol: n(A) = n(B)

Example: A = { 1, 5, 7 } , B = { 2, 3, 4 }

n(A) = n(B) = 3

So A is equivalent to B.

Chapter 1 Section 1 - Slide 22Copyright © 2009 Pearson Education, Inc.

Empty (or Null) Set

The null set (or empty set ) contains absolutely NO elements.

Symbol: or

Chapter 1 Section 1 - Slide 23Copyright © 2009 Pearson Education, Inc.

Universal Set

The universal set contains all of the possible elements which could be discussed in a particular problem.

Symbol: U

Chapter 1 Section 1 - Slide 24Copyright © 2009 Pearson Education, Inc.

Section 2.2

Subsets

Chapter 1 Section 1 - Slide 25Copyright © 2009 Pearson Education, Inc.

Subsets

A set is a subset of a given set if and only if all elements of the subset are also elements of the given set.

Symbol:

To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B. The symbol for “not a subset of” is .

Chapter 1 Section 1 - Slide 26Copyright © 2009 Pearson Education, Inc.

Example: Determine whether set A is a subset of set B.

A = { 3, 5, 6, 8 }B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Solution: All of the elements of set A are contained in set

B, so

Determining Subsets

A B .

Chapter 1 Section 1 - Slide 27Copyright © 2009 Pearson Education, Inc.

Proper Subset

All subsets are proper subsets except the subset containing all of the given elements, that is, the given set must contain one element not in the subset (the two sets cannot be equal).

Symbol:

Chapter 1 Section 1 - Slide 28Copyright © 2009 Pearson Education, Inc.

Determining Proper Subsets

Example: Determine whether set A is a proper subset of

set B.A = { dog, cat }B = { dog, cat, bird, fish }

Solution: All the elements of set A are contained in set B,

and sets A and B are not equal, therefore A B.

Chapter 1 Section 1 - Slide 29Copyright © 2009 Pearson Education, Inc.

Determining Proper Subsets (continued)

Example: Determine whether set A is a proper subset of

set B.A = { dog, bird, fish, cat }B = { dog, cat, bird, fish }

Solution: All the elements of set A are contained in set B,

but sets A and B are equal, therefore A B.

Chapter 1 Section 1 - Slide 30Copyright © 2009 Pearson Education, Inc.

Number of Distinct Subsets

The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set A.

Example: Determine the number of distinct subsets

for the given set { t , a , p , e }. List all the distinct subsets for the given set:

{ t , a , p , e }.

Chapter 1 Section 1 - Slide 31Copyright © 2009 Pearson Education, Inc.

Solution: Since there are 4 elements in the given set,

the number of distinct subsets is

24 = 2 • 2 • 2 • 2 = 16 subsets. {t,a,p,e},

{t,a,p}, {t,a,e}, {t,p,e}, {a,p,e},

{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e},

{t}, {a}, {p}, {e}, { }

Number of Distinct Subsets continued