section 3.1 statements and logical connectives · pdf filestatements and logical connectives...

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Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 3.1

Statements and Logical Connectives


What You Will Learn - LogicStatements, quantifiers, and compound statements

Statements involving the words not, and, or, if … then, and if and only if

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.

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MathematiciansGottfried 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.

Charles Dodgson, better known as Lewis Carroll, incorporated many interesting ideas from logic into his books.

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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

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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 and can

be assigned a truth value.

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

Be careful when negating statements that contain quantifiers.

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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.

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None are.

Some are not.

All are.

Some are.


Example 1: Write Negations

Write the negation of the statement.

Some telephones can take photographs.

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Example 1: Write NegationsWrite the negation of the statement. (at least two ways)

All houses have two stories.

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Compound StatementsStatements 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.

The symbol used in logic to show the negation of

a statement is ~. It is read “not”.

The negation of p is: ~ p.

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And Statements (Conjunction)

⋀ is the symbol for a conjunction and

is read “and.” p ⋀ q.

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Write the following conjunction in symbolic form:

Green Day is not on tour,

but Green Day is recording a new CD.


Or Statements (Disjunction)The disjunction is symbolized by ⋁Let

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

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

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

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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 dayWrite the statement in symbolic form.Dinner includes soup, and salad or

vegetable of the day.

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Let p: Dinner includes soup.q: Dinner includes salad.r: Dinner includes the vegetable of

the dayWrite the statement in symbolic form.Dinner includes soup, and salad or


Write the statement in symbolic form.Dinner includes soup and salad, or


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Example 5:Let p: The house is for sale.

q: We can afford to buy the house.Write the symbolic statement in words.

p ⋀ ~q

~p ⋁ ~q

~(p ⋀ q)

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If-Then Statements

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.

If p, then q is symbolized as: p → q.

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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.

Solutionp → q

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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.

Solutionq → ~p

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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 → p)

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If and Only If Statements

The biconditional is symbolized by ↔

and is read “if and only if.”

If and only if is sometimes abbreviated

as “iff.”

The statement p ↔ q is read “p if and

only if q.”

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Example 8: Write Statements Using the BiconditionalLetp: Alex plays goalie on the lacrosse team.q: The Titans win the Champion’s Cup.Write the symbolical statement in words.

p ↔ qSolutionAlex plays goalie on the lacrosse team if

and only if the Titans win the Champion’s Cup.

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q ↔ ~pSolutionThe Titans win the Champion’s cup if and

only if Alex does not play goalie on the lacrosse team.

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~(p ↔ ~q)SolutionIt is false that Alex plays goalie on the

lacrosse team if and only if the Titans do not win the Champion’s Cup.

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Logical Connectives

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Section 3.2

Truth Tables for Negation, Conjunction,

and Disjunction


Negation Truth Table

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p ~p

Case 1 T F

Case 2 F T

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.

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p q p ⋀ q

Case 1 T T T

Case 2 T F F

Case 3 F T F

Case 4 F F F

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Disjunction Truth Table

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

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p q p ⋁ q

Case 1 T T T

Case 2 T F T

Case 3 F T T

Case 4 F F F


Example 3: Truth Table with a Negation

Construct a truth table for ~(~q ⋁ p).

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Example 7: Use the Alternative Method to Construct a Truth Table

Construct a truth table for ~p ⋀ ~q.

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Example 9: 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.

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Example 9: 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.

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.

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Section 3.3

Truth Tables for the

Conditional and

Biconditional

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Conditional

The conditional statement p → q is true in every case except when p is a true statement and q is a false statement.

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p q p → q

Case 1 T T T

Case 2 T F F

Case 3 F T T

Case 4 F F T


This leads to some “weird” stuffIf you are in chemistry class right now, then

you have green skin.

Since none of you are in chemistry class

right now, the entire statement is true!

Note: Difference between statement being

true and you having green skin.

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Construct a truth table for the statement ~p → ~q.

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Biconditional

The biconditional statement, p ↔ q is true only when p and q have the same truth value, that is, when both are true or both are false.

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Construct a truth table for the statement ~p ↔ (~q → r).

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The graph on the next slide represents the student population by age group in 2009 for the State College of Florida (SCF). Determine the truth value of the following statements.

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1. If 37% of the SCF population is younger than 21 or 26% of the SCF population is age 21–30, then 13% of the SCF population is age 31–40.

2. 3% of the SCF population is older than 50 and 8% of the SCF population is age 41–50, if and only if 19% of the SCF population is age 21–30.

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Self-Contradiction

A self-contradiction is a compound statement

that is always false.

A tautology is a compound statement that is

always true.

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Construct a truth table for the statement (p ↔ q) ⋀ (p ↔ ~q).

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Construct a truth table for the statement (p ⋀ q) → (p ⋁ r).

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An implication is a conditional

statement that is a tautology.

Determine whether the conditional

statement [(p ⋀ q) ⋀ q] → q is an

implication.

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Section 3.4

Equivalent Statements


Equivalent Statements

Two statements are equivalent,symbolized ⇔ or ≡, if both statements have exactly the same truth values in the answer columns of the truth tables.Sometimes the words logically equivalent are used in place of the word equivalent.

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Are the following equivalent?p ⋀ (q ⋁ r) (p ⋀ q) ⋁ (p ⋀ r)

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Determine which statement is logically equivalent to “it is not true that the tire is both out of balance and flat.”a) if the tire is not flat, then the tire is not out of balance.b) the tire is not out of balance or the tire is not flat.c) the tire is not flat and the tire is not out of balance.d) if the tire is not out of balance, then the tire is not flat.p: The tire is out of balance q: The tire is flat.

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De Morgan’s Laws

~ (p ∧ q) ⇔ ~ p ∨ ~ q

~ (p ∨ q) ⇔ ~ p ∧ ~ q

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Example 5: Using De Morgan’s Laws to Write an Equivalent Statement

Write a statement that is logically equivalent to “It is not true that tomatoes are poisonous or eating peppers cures the common cold.”

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To change a conditional statement into a disjunction, negate the antecedent, change the conditional symbol to a disjunction symbol, and keep the consequent the same.

The Conditional Statement Written as a Disjunction

p→ q ⇔ ~p ∨ q

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Write a conditional statement that is logically equivalent to “The Oregon Ducks will win or the Oregon State Beavers will lose.” Assume that the negation of winning is losing.

Example 7: Rewriting a Disjunction as a Conditional Statement

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The Negation of the Conditional Statement Written as a Conjunction

~(p → q) ≡ p ⋀ ~q

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Example 9: Write an Equivalent Statement

Write a statement that is equivalent to

“It is false that if you hang the picture

then it will be crooked.”

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Variations of the Conditional Statement

The variations of conditional

statements are the converse of the

conditional, the inverse of the

conditional, and the contrapositive of

the conditional.

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Variations of the Conditional Statement

Name Symbolic Form

Read

Conditional p → q “If p, then q”

Converse q → p “If q, then p”

Inverse ~p → ~q “If not p, then not q”

Contrapositive ~q → ~p “If not q, then not p”

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Example 10: The Converse, Inverse, and Contrapositive

For the conditional statement “If the song contains sitar music, then the song was written by George Harrison,” write thea) converse.b) inverse.c) contrapositive.

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Section 3.6

Euler Diagrams

and Syllogistic Arguments

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Syllogistic Arguments

Another form of argument is called a

syllogistic argument, better known as

syllogism.

The validity of a syllogistic argument is

determined by using Euler

(pronounced “oiler”) diagrams.

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Symbolic Arguments Versus Syllogistic Arguments

Euler diagramsall are, some are, none are, some are not

Syllogistic argument

Truth tables or by comparison with standard forms of arguments

and, or, not, if-then, if and only if

Symbolic argument

Methods of determining validity

Words or phrases used

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Example 3: Ballerinas and Athletes

Determine whether the following syllogism is valid or invalid.

All ballerinas are athletic.

Keyshawn is athletic.

∴ Keyshawn is a ballerina.

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Example 3: Ballerinas and AthletesSolution

Keyshawn is athletic, so must be placed in the set of athletic people, which is A. We have a choice, as shown above.

All ballerinas, B, are athletic, A.

The conclusion does not necessarily follow from the set of premises. The argument is invalid.

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Example 4: Parrots and Chickens


No parrots eat chicken.

Fletch does not eat chicken.

∴ Fletch is a parrot.

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Example 4: Parrots and ChickensSolution

The first premise tells us that parrots and things that eat chicken are disjoint sets—that is, sets that do not intersect.

Fletch is not a parrot, the argument is invalid, or is a fallacy.

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Example 5: A Syllogism Involving the Word Some


All As are Bs.

Some Bs are Cs.

∴ Some As are Cs.

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Example 6: Fish and Cows


No fish are mammals.

All cows are mammals.

∴ No fish are cows.

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Example 6: Fish and CowsSolution

The first premise tells us that fish and mammals are disjoint sets. The second tells us that the set of cows is a subset of the set of mammals.The conclusion necessarily follows from the premises and the argument is valid.

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Section 3.7

Switching Circuits


Series Circuit

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Parallel Circuit

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Example 2: Representing a Switching Circuit with Symbolic Statements

a.Write a symbolic statement that represents the circuit.

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Example 2: Representing a Switching Circuit with Symbolic Statements

b.Construct a truth table to determine when the light will be on.

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Example 2: Representing a Switching Circuit with Symbolic Statements Solution

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Example 3: Representing a Symbolic Statement as a Switching CircuitDraw a switching circuit that represents

[(p ⋀ ~q) ⋁ (r ⋁ q)] ⋀ s.

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Equivalent Circuits

Equivalent circuits are two circuits

that have equivalent corresponding

symbolic statements.

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Equivalent Circuits

Sometimes two circuits that look very

different will actually have the exact

same conditions under which the light

will be on.

The truth tables have identical answer

columns.

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Example 4: Are the Circuits Equivalent?

Determine whether the two circuits are equivalent.

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p ⋁ (q ⋀ r)

(p ⋁ q) ⋀ (p ⋁ r)

Solution

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The answer columns are identical.

Solution

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Therefore, p ⋁ (q ⋀ r) is equivalent to (p ⋁ q) ⋀ (p ⋁ r) and the two circuits

are equivalent.

Solution

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section 3.1 statements and logical connectives · pdf filestatements and logical connectives...

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