Set Theory and Logic

Foundations 30Mrs. Wirz

3.1 Types of Sets and Set Notation

Introduction – Venn Diagrams

Students that play sports Students that have a part-time job

All students in Mrs. Wirz’s class

3.1 Types of Sets and Set NotationTerminology

Set: A collection of distinguishable objects for example, the set of whole numbers is w={1,2,3,…}

Element: An object in a set, for example 3 is an element of D, the set of all digits.

Universal Set: The set of all elements being considered, for example the universal set of digits is D={1,2,3,4,5,6,7,8,9}

Subset: A set whose elements all belong to another set, for example, the set of odd digits, O={1,3,5,7,9} is a subset of D.

In set notation this is written as O D.

3.1 Types of Sets and Set NotationTerminology

Complement: All the elements that belong to a universal set, but do not belong to a subset.For example, the complement of subset O is O’={2,4,6,8}, which happens to be all the even digits.

Empty set: A set with no elements. The set is denoted as { } or .

Disjoint: Two or more sets having no elements in common; for example, the set of even numbers and the set of odd numbers are disjoint.

3.1 Types of Sets and Set Notation

A. List the elements of the universal set of students in Foundations 30.

3.1 Types of Sets and Set Notation

3.1 Types of Sets and Set Notation

F

S W

3.1 Types of Sets and Set Notation

F is the universal set. W and S are subsets of F.

Write W in set notation.

W’ is the complement of W Describe what W’ contains Write W’ in set notation

What would make an empty set?

3.1 Types of Sets and Set NotationSet Notation

3.1 Types of Sets and Set NotationExamples

Example 1:A. Using set notation, define the

universal set, S, of all natural numbers 1 to 500.

B. Using set notation define, the subsets, F and T, which indicate the multiples of 5 and 10, from 1 to 500.

C. Represents the sets and subsets using a Venn diagram.

3.1 Types of Sets and Set NotationExamples

Complete example 2 and 3 on handout.

3.1 Types of Sets and Set NotationTerminology

Finite Set: A set with a countable number of elements; for example, the set of even numbers less than 10, E+{2,4,6,8} is finite.

Infinite Set: A set with infinite number of elements; for example the set of natural numbers, N={1, 2, 3,…}

Mutually Exclusive: Two or more events that cannot occur at the same time; for example, the Sun rising and the Sun setting are mutually exclusive events.

3.1 Types of Sets and Set NotationThings you should also know

Sets are equal if they contain exactly the same elements, even if the elements are listed in different orders.

You may not be able to count all the elements in a very large or infinite set, such as the set of real numbers.

The sum of the number of elements in a set and its complement is equal to the number of elements in the universal set

When two sets A and B are disjoint,

3.1 Types of Sets and Set NotationAssignment

Text: page 154-157Do # 2,5,8,11

3.2 Exploring Relationships between Sets

Sets that are not disjoint share common elements.

Each area of a Venn diagram represents something different.

When two non-disjoint sets are represented in a Venn diagram, you can count the elements in both sets by counting the elements in each region only once.

3.2 Exploring Relationships between Sets

3.2 Exploring Relationships between Sets

Each element in a universal set appears only once in a Venn diagram.

If an element occurs in more than one set, it is placed in the area of the Venn diagram where the sets overlap.

3.2 Exploring Relationships between Sets - Assignment

Page 160-161 (on the back of this handout)

# 1, 3

3.3 Intersection and Union of Two Sets - Notation

In set notation, is read as “intersection of A and B.” It denotes the elements that are common to both A and B. The intersection is the region where two sets overlap in the Venn diagram.U

AB

3.3 Intersection and Union of Two Sets - Notation

is read as “union of A and B.” It denotes all elements that belong to at least one of A or B.

U

A B

3.3 Intersection and Union of Two Sets - Notation

A\B is read as “ A minus B.” It denotes the set of elements that are in set A but not in set B.

Do Example 1

3.3 Intersection and Union of Two Sets – Principle of Inclusion and Exclusion

The number of elements in the union of two sets is equal to the sum of the number of elements in each set, less the number of elements in both sets.

Do example 2

3.3 Intersection and Union of Two Sets – summary

3.3 Intersection and Union of Two Sets – summary

3.3 Intersection and Union of Two Sets – Assignment

http://www.youtube.com/watch?v=s8H-fMhK2g0

Page 172- 174, #1, 6,8,11

3.4 Applications of Set Theory – When is set theory ever used?

Computer Science• Software• Internet searches• Networking• Programing

Electrical Engineering Geometry and Topology Biology, Physics, and Chemistry

3.4 Applications of Set Theory Examples and Assignment

Do examples “Investigate the Math,” example 1,2, and 3.

Do Questions #2,4,6, and 9

3.5 Conditional Statements and Their Converse - Terminology

Conditional Statement: An “if-then” statement.

example: “If it is raining outside, then we practice indoors.”Hypothesis: “It is raining outside.”Conclusion: “We practice indoors.”The hypothesis is the statement that follows “if,” and the conclusion follows “then.”

3.5 Conditional Statements and Their Converse – Determining if a Conditional Statement is True or False

Case 1: It rains outside, and we practice indoors.When the hypothesis and conclusion are both true, the conditional statement is true. Case 2: It does not rain outside, and we practice

indoors.When the hypothesis and the conclusion are both false, then the conditional statement is true. Case 3: It does not rain outside, and we practice

outdoors.When the hypothesis is false and the conclusion is true, the conditional statement is true. Case 4: It rains outdoors, and we practice outdoors.When the hypothesis is true and the conclusion is false, then the conditional statement is false.

3.5 Conditional Statements and Their Converse – Truth Table

Let p represent the hypothesis and let q represent the conclusion.

p q pq

True True True

False False True

False True True

True False False

3.5 Conditional Statements and Their Converse - Terminology

Converse: A conditional statement in which the hypothesis and the conclusion are switched.

example: If the conditional statement is:If it is raining outside, then we will practice indoors.The converse would be: If we practice indoors, then it is raining outside.A counter-example could disprove this statement. Such as, you could be practicing indoors because the field is being used or under repair.

3.5 Conditional Statements and Their Converse - Terminology

Biconditional: A biconditional statement is a conditional statement whose converse is also true. A biconditional statement is written as “p if and only if q.”

Example: conditional statement: If a glass is half-empty, then it is

half-full.converse: If a glass is half-full, then it is half-empty.

Both the conditional statement and converse are true, therefore the biconditional statement is:A glass is half-empty if and only it is half-full.

Notation for biconditional: pq

3.5 Conditional Statements and Their Converse - Assignment

Read over the “in summary.” Do questions 2, 3, 4, 5, 6, 10, and 13

from pages 204-205

3.6 The Inverse and the Contrapositive of Conditional Statements - Terminology

Inverse: A statement that is formed by negating both the hypothesis and the conclusion of a conditional statement;

for example, for the statement “If a number is even then it is divisible by 2,” the inverse is “If a number is not even, then it is not divisible by 2.”

3.6 The Inverse and the Contrapositive of Conditional Statements - Terminology

Contrapositive: A statement that is formed by negating both the hypothesis and the conclusion of the converse of a conditional statement;

For example, for the statement “If a number is even, then it is divisible by 2,” the contrapositive is “If a number is not divisible by 2, then it is not even.”

3.6 The Inverse and the Contrapositive of Conditional Statements - Example

Mrs. Wirz said, “If a polygon is a triangle then it has three sides.”

A. Is this statement true? Explain.

B. Write the converse of this statement. Is it true? Explain.

C. Write the inverse of this statement. Is it true? Explain.

D. Write the contrapositive of this statement. Is it true? Explain.

3.6 The Inverse and the Contrapositive of Conditional Statements - Example

Consider the statement, “If you live in Saskatoon, then you live in Saskatchewan.”

A. Is this statement true? Explain.

B. Write the converse of this statement. Is it true? Explain.

C. Write the inverse of this statement. Is it true? Explain.

D. Write the contrapositive of this statement. Is it true? Explain.

3.6 The Inverse and Contrapositive

If a conditional statement is true, then its contrapositive is true, and vice versa.

If the inverse of a conditional statement is true, then the converse of the statement is also true, and vice versa.

3.6 The Inverse and Contrapositive

Examples – Read to understand examples 1 and 2. Pay special attention to the notation boxes. Complete the Your Turns for example 1 and 2.

Assignment - #1-8, page 214-215

Chapter 3: Set Theory and Logic

Review Assignmentpage 220, #1,2,3,4,6,8.

Due: Monday September 24

Test – Tuesday September 25