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Section 3.1

Statements and Logical Connectives

What You Will Learn

Statements, quantifiers, and compound statements

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.

Mathematicians

George Boole (1815 - 1864) is said to be the founder of symbolic logic because he had such impressive

work in this area. Charles Dodgson, better known as Lewis Carroll,((1832-1898)-some well known works of his include-

Alice in wonderland and Through the Looking Glass) incorporated many interesting ideas from logic into

his books.

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.

Statements and Logical Connectives

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 and can be assigned a truth value.

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

Be careful when negating statements that contain quantifiers.

Negation of Quantified Statements

Form of statement and its negation

All are.>>>some are not

None are.>>>>>>some are

Some are.>>>>>>none are

Some are not.>>>>>>all are

Negation of Quantified Statements

Example 1: Write Negations

Write the negation of the statement.

Some telephones can take photographs.

Example 1: Write Negations

Write the negation of the statement.

All houses have two stories.

Compound Statements

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

Not Statements (Negation)

The symbol used in logic to show the negation of a statement is Ε. It is read ͞not".

The negation of p is: ~ p.

And Statements (Conjunction)

The conjunction of p and q is: p ٿ

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

Example 2: Write a Conjunction

Write the following conjunction in symbolic form:

Green Day is not on tour, but Green Day is recording a new CD.

Example 2: Write a Conjunction

Solution

Let t and r represent the simple statements.

t: Green Day is on tour. r: Green Day is recording a new CD. In symbolic form, the compound statement is ~t ٿ

Or Statements (Disjunction)

The disjunction is symbolized by ڀ

In this book the ͞or" will be the inclusive or (except where indicated in the exercise set).

The disjunction of p and q is: p ڀ

Example 3: Write a Disjunction

Let p: Maria will go to the circus. q: Maria will go to the zoo.

Write the statement in symbolic form.

Maria will go to the circus or Maria will go the zoo.

Solution

p ڀ

Example 3: Write a Disjunction

Let/Given

p: Maria will go to the circus. q: Maria will go to the zoo.

Write the statement in symbolic form.

Maria will go to the zoo or Maria will not go the circus.

Solution

q ڀ

Compound Statements

When a compound statement contains more than one connective, a comma can be used to indicate which simple statements are to be grouped together. When we write the compound statement symbolically, the simple statements on the same side of the comma are to be grouped together within parentheses. Example 4: Understand How Commas Are Used to Group Statements

Let p: Dinner includes soup.

q: Dinner includes salad. r: Dinner includes the vegetable of the day

Write the statement in symbolic form.

Dinner includes soup, and salad or vegetable of the day.

Solution p ٿ (q ڀ

Example 4: Understand How Commas Are Used to Group Statements

Let p: Dinner includes soup.

q: Dinner includes salad. r: Dinner includes the vegetable of the day

Write the statement in symbolic form.

Dinner includes soup and salad, or vegetable of the day.

Solution (p ٿ q) ڀ

Example 5: Change Symbolic Statements into Words

Let p: The house is for sale.

q: We can afford to buy the house.

Write the symbolic statement in words.

p ٿ

Solution

The house is for sale and we cannot afford to buy the house.

Example 5: Change Symbolic Statements into Words

Let p: The house is for sale.

q: We can afford to buy the house.

Write the symbolic statement in words.

~p ڀ

Solution

The house is not for sale or we cannot afford to buy the house.

Example 5: Change Symbolic Statements into Words

Let p: The house is for sale.

q: We can afford to buy the house.

Write the symbolic statement in words.

~(p ٿ

Solution

It is false that the house is for sale and we can afford to buy the house.

If-Then Statements

The conditional is symbolized by ї 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.

If p, then q is symbolized as: p ї q.

Example 6: Write Conditional Statements

Let p: The portrait is a pastel.

q: The portrait is by Beth Anderson.

Write the statement symbolically.

If the portrait is a pastel, then the portrait is by Beth Anderson.

Solution

p ї q

Example 6: Write Conditional Statements

Let p: The portrait is pastel.

q: The portrait is by Beth Anderson.

Write the statement symbolically.

If the portrait is by Beth Anderson, then the portrait is not a pastel.

Solution

q ї Εp

Example 6: Write Conditional Statements

Let p: The portrait is pastel.

q: The portrait is by Beth Anderson.

Write the statement symbolically.

It is false that if the portrait is by Beth Anderson, then the portrait is a pastel.

Solution

~(q ї p)

If and Only If Statements

The biconditional is symbolized by ў and is read ͞if and only if." If and only if is sometimes abbreǀiated as ͞iff." The statement p ў q is read ͞p if and only if q." Example 8: Write Statements Using the Biconditional Let p: Alex plays goalie on the lacrosse team. q: The Titans win the Champion's Cup.

Write the symbolical statement in words.

p ў q

Solution

Aledž plays goalie on the lacrosse team if and only if the Titans win the Champion's Cup. Example 8: Write Statements Using the Biconditional Let p: Alex plays goalie on the lacrosse team. q: The Titans win the Champion's Cup.

Write the symbolical statement in words.

q ў Εp

Solution

The Titans win the Champion's cup if and only if Aledž does not play goalie on the lacrosse team. Example 8: Write Statements Using the Biconditional Let p: Alex plays goalie on the lacrosse team. q: The Titans win the Champion's Cup.

Write the symbolical statement in words.

~(p ў Εq)

Solution

It is false that Alex plays goalie on the lacrosse team if and only if the Titans do not win the Champion's

Cup.

Logical Connectives

Section 3.2

Truth Tables for Negation, Conjunction, and Disjunction

What You Will Learn

Truth tables for negations, conjunctions, and disjunctions

Truth Table

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

Negation Truth Table

p ~p

Case 1 T F

Case 2 F T

Compound Statement Truth Table

p q

Case 1 T T

Case 2 T F

Case 3 F T

Case 4 F F

Conjunction Truth Table: The conjunction is true only when both p and q are true. p q p ٿ

Case 1 T T T

Case 2 T F F

Case 3 F T F

Case 4 F F F

Disjunction Truth Table:The disjunction is true when either p is true, q is true, or both p and q are true.

p q p ڀ

Case 1 T T T

Case 2 T F T

Case 3 F T T

Case 4 F F F

Negation

Negation ~p is read ͞not p."

If p is true, then ~p is false;

if p is false, then ~p is true. In other words, ~p will always have the opposite truth value of p.

Conjunction

Conjunction p ٿ

p ٿ

Disjunction

Disjunction p ڀ

p ڀ

In other words, p ڀ

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

ї if it is conditional

ў if it is biconditional

Constructing Truth Tables

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 pair first.

Constructing Truth Tables

3. Complete the column under the connective within the parentheses, if present. You will use the

truth values of the connective in determining the final answer in step 5.

Constructing Truth Tables

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

Constructing Truth Tables

5. Complete the column under any remaining connectives. The 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.

Constructing Truth Tables

5. (continued)

For a negation, negate the values of the last column completed within the grouping symbols on the right of the negation. Circle or highlight the answer column and number the columns in the order they were completed.

Example : Truth Table with a Negation

Construct a truth table for ~(~q ڀ

Example: Construct a Truth Table

Construct a truth table for ~p ٿ

Example: Determine the Truth Value of a Compound Statement Determine the truth value for each simple statement. Then, using these truth values, determine the truth value of the compound statement.

15 is less than or equal to 9.

Example : Determine the Truth Value of a Compound Statement Let p: 15 is less than 9. q: 15 is equal to 9.

Both p and q are false.

p ڀ

F ڀ

F Example : Determine the Truth Value of a Compound Statement Determine the truth value for each simple statement. Then, using these truth values, determine the truth value of the compound statement. George Washington was the first U.S. president or Abraham Lincoln was the second U.S. president, but there has not been a U.S. president born in Antarctica. Solution to Example : Determine the Truth Value of a Compound Statement Let p: George Washington was the first U.S. president. q: Abraham Lincoln was the second U.S. president. r: There has been a U.S. president who was born in Antarctica. The statement can be written in symbolic form as (p ڀ q) ٿ p is true, q is false, r is false.

Since r is false, ~r is true.

The statement is (p ڀ q) ٿ

p is true, q is false, ~r is true. (p ڀ q) ٿ (T ڀ F) ٿ

T ٿ

T

The original compound statement is true.

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