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

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

AND

Copyright © 2009 Pearson Education, Inc. Chapter 3 Section 1 - Slide 2

Chapter 3

Logic


WHAT YOU WILL LEARN• Statements, quantifiers, and

compound statements• Statements involving the words not,

and, or, if… then…, and if and only if• Truth tables for negations,

conjunctions, disjunctions, conditional statements, and biconditional statements


Section 1

Statements and Logical Connectives


HISTORY—The Greeks:

Aristotelian logic: The ancient Greeks were the first people to look at the way humans think and draw conclusions. Aristotle (384-322 B.C.) is called the father of logic. This logic has been taught and studied for more than 2000 years.


Mathematicians

Gottfried Wilhelm Leibniz (1646-1716) believed that all mathematical and scientific concepts could be derived from logic. He was the first to seriously study symbolic logic. In this type of logic, written statements use symbols and letters.

George Boole (1815 – 1864) is said to be the founder of symbolic logic because he had such impressive work in this area.


Logic and the English Language

Connectives - words such as and, or, if, then Exclusive or - one or the other of the given

events can happen, but not both. Inclusive or - one or the other or both of the

given events can happen.


Statements and Logical Connectives

Statement - A sentence that can be judged either true or false. Labeling a statement true or false is called

assigning a truth value to the statement. Simple Statements - A sentence that conveys

only one idea and can be assigned a truth value.

Compound Statements - Sentences that combine two or more simple statements.


Negation of a Statement

Negation of a statement – change a statement to its opposite meaning.

The negation of a false statement is always a true statement.

The negation of a true statement is always a false statement.


Quantifiers

Quantifiers - words such as all, none, no, some, etc…

Be careful when negating statements that contain quantifiers.


Negation of Quantified Statements

Form of statement

All are.

None are.

Some are.

Some are not.

Form of negation

Some are not.

Some are.

None are.

All are.

None are.

Some are not.

All are.

Some are.


Example: Write Negations

Write the negation of the statement.

Some candy bars contain nuts.

Solution: Since some means “at least one” this statement is true. The negation is “No candy bars contain nuts,” which is a false statement.


Example: Write Negations continued

Write the negation of the statement.

All tables are oval.

Solution: This is a false statement since some tables are round, rectangular, or other shapes. The negation would be “Some tables are not oval.”


Compound Statements

Statements consisting of two or more simple statements are called compound statements.

The connectives often used to join two simple statements are and, or, if…then…, and if and only if.


Not Statements (Negation)

The symbol used in logic to show the negation of a statement is ~. It is read “not”.

Example: The negation of p is: ~ p.


And Statements (Conjunction)

is the symbol for a conjunction and is read “and.”

The other words that may be used to express a conjunction are: but, however, and nevertheless.

Example: The conjunction of p and q is: p ^ q.


Example: Write a Conjunction

Write the following conjunction in symbolic form:The dog is gray, but the dog is not old.

Solution:

Let p and q represent the simple statements.

p: The dog is gray.

q: The dog is old.

In symbolic form, the compound statement is

p Λ ~ q


Or Statements (Disjunction)


Example: Write a Disjunction

Write the statement in symbolic form. Carl will not go to the movies or Carl will not go to the baseball game.

~ ~p q

Solution:

Let p and q represent the simple statements.

p: Carl will go to the movies.

q: Carl will go to the baseball game.

In symbolic form, the compound statement is


If-Then Statements (continued)

The conditional is symbolized by and is read “if-then.”

The antecedent is the part of the statement that comes before the arrow.

The consequent is the part that follows the arrow.

Example: If p, then q is symbolized as: p q.


Example: Write a Conditional Statement

Let p: Nathan goes to the park.

q: Nathan will swing.

Write the following statements symbolically.

a. If Nathan goes to the park, then he will swing.

b. If Nathan does not go to the park, then he will not swing.

Solutionsa) p q b) ~ ~p q


If and Only If Statements (Biconditional)

The biconditional is symbolized by and is read “if and only if.”

If and only if is sometimes abbreviated as “iff.”

The biconditional p q is the conjunction of the two conditionals p q and q p:

p q = (p q) (q p)


Example: Write a Statement Using the Biconditional

Let p: The dryer is running.

q: There are clothes in the dryer.

Write the following symbolic statements in words.

a) b)

Solutions:

a. There are clothes in the dryer if and only if the dryer is running.

b. It is false that the dryer is running if and only if there are no clothes in the dryer.

q p


Section 2

Truth Tables for Negation,Conjunction, and Disjunction


Truth Table

A truth table is used to determine when a compound statement is true or false.


Negation Truth Table

TCase 2 F

FCase 1 T

~p p


Conjunction Truth Table

The conjunction is true only when both p and q are true.

FFFCase 4

FTFCase 3

FFTCase 2

TTTCase 1

qp p q


Disjunction

The disjunction is true when either p is true, q is true, or both p and q are true.

FFFCase 4

TTFCase 3

TFTCase 2

TTTCase 1

qp p q


General Procedure for Constructing Truth Tables

1. Determine if the statement is a negation, conjunction, disjunction, conditional, or biconditional.

The answer to the truth table appears under:

~ if it is a negation

B if it is a conjunction

O if it is a disjunction

if it is conditional

if it is biconditional


General Procedure for Constructing Truth Tables (continued)

2. Complete the columns under the simple statements, p, q, r, and their negations ~p, ~q, ~r, within parentheses, if present. If there are nested parentheses work with the innermost first.

3. Complete the column under the connective within parentheses, if present. You will use the truth values of the connective in determining the final answer in step 5.

4. Complete the column under any remaining statements and their negation.


General Procedure for Constructing Truth Tables (continued)

5. Complete the column under any remaining connectives. Answer will appear under the column determined in step 1.

For a conjunction, disjunction, conditional or biconditional, obtain the value using the last column completed on the left side and on the right side of the connective.

For a negation, negate the values of the last column completed within the grouping symbols on the right of the negation. Number the columns in the order they were completed.


Example: Truth Table with a Negation and Conjunction

Construct a truth table for ~(~q p).Solution: Construct a standard four case truth

table.

p q ~ (~q B p)

TTFF

TFTF

TFTT

FTFT

FTFF

TTFF

Then fill-in the table in order, as follows:

False only when p is true and q is false.2314

Ù

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

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