1 intro to logic

© Greenville County Schools© Greenville County Schools

Introduction to LogicSimple and Compound Statements

Connectives NOT, AND, OR

Resources: Resources: HRW Geometry, Lesson 12.2HRW Geometry, Lesson 12.2


Introduction Instruction Examples Practice

How can you tell How can you tell when a complicated when a complicated statement is true or statement is true or false?false?

In the nineteenth century, In the nineteenth century, George Boole George Boole symbolic symbolic logiclogic. He believed logical . He believed logical ideas could be calculated ideas could be calculated symbolically. His methods symbolically. His methods allow us to perform allow us to perform calculations to decide if calculations to decide if statements are true or statements are true or false and whether logical false and whether logical arguments are valid.arguments are valid.



Please go back or choose a topic from above.


List of Instructional Pages

1. Statements2. Simple and Compound

Statements3. The Connectives: NOT, AND, OR4. The Connectives:

Negation5. Negation: Truth Table6. The Connectives:

Conjunction7. Conjunction: Truth

Table

8. The Connectives: Disjunction

9. Disjunction: Truth Table

10. Truth Tables: Negating a Conjunction

11. Truth Tables: The Disjunction of Two Negations

12. Truth Tables: Truth Tables: Logically Equivalent Statements



A statementstatement in is a declarative sentence that is either true or false. We represent statements by lowercase letters such as p, q, or r.

Examples of statements:

• AIDS is a leading killer of AIDS is a leading killer of women.women.

• If you eat less and exercise If you eat less and exercise more, you will lose weight.more, you will lose weight.

• Cebu is in Luzon.Cebu is in Luzon.

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These are not not statements:

• Beam me up Scotty.

• When did dinosaurs become extinct?

• This statement is false.

Introduction to Symbolic Logic


Identify which of the following sentences are “statements” as defined in logic.

YES

NO

YES

NO

YES

YES

NO

Kimberly lives in Cebu City.

Drop that puppy right now!

My pencil is broken.

Do we have an assignment for today?

The Philippines has more than 8 000 islands.

The complement of 50° is 40°.

Please hand me my bag.

Is this a logical “statement”?


A Compound StatementA Compound Statement


A simple statementsimple statement contains a single idea.


A Simple StatementA Simple Statement

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“The 1998 Yankees were the best team in the history of baseball.”

A compound compound statementstatement contains several ideas combined together.

If you break your lease, then you forfeit your deposit.

Introduction to Symbolic Logic



Words used to join the ideas of a compound statement are called connectives.connectives. Three of the connectives are not, and, and or.


The SymbolsThe Symbols

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NOT ~AND

OR

∧∨

NegationNegation NOT

ConjunctionConjunction AND

DisjunctionDisjunction OR



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A negationnegation is a statement expressing the idea that something is not true. We represent negation by the symbol ~ and use the word “notnot”.

If p represents “The blue whale is the largest living creature,” then

~p represents “The blue whale is notnot the largest living creature.”

The Connectives - Negation



Consider the following statements:p: The sun is a star.~p: The sun is not a star.When the first statement, p, is true, the second statement, ~p, is false, and vice versa. We can represent this in a truth table.


Graphic

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NegationNegationThis is where the symbol This is where the symbol that represents your first that represents your first statement will go.statement will go.

This is where the symbol This is where the symbol that represents your that represents your second statement will go.second statement will go.

Since the first entry is Since the first entry is true, the second entry true, the second entry is false.is false.

Since we are looking Since we are looking at the at the negationnegation of the of the statement, here we statement, here we need the need the oppositeopposite of of the previous columnthe previous column

pp

TT

FF

~p~p

FF

TT

By convention this first By convention this first entry is usually TRUE.entry is usually TRUE.


Give the truth value of the given statements, its negation and the negation’s truth value.

StatementTruth Value Negation

The penguin is classifiedas a bird.

T The penguin is not classifiedas a bird.

F

Truth Value

The Philippine flag has five colors.

F The Philippine flag does not have five colors.

T

The difference of 38 and 13 is not equal to 25.

F The difference of 38 and 13 is equal to 25.

T

Mars is not the hottest planet in the solar system.

T Mars is the hottest planet in the solar system.

F

Two points are always collinear.

T Two points are not always collinear.

F

Two planes does not intersect at a point.

T Two planes intersect at a point.

F


A conjunctionconjunction expresses the idea of and. We use the symbol Λ to represent a conjunction.

p :

d :

p Λ d :

~p Λ ~d : Now write the conjunction of the negations of each statement:

NOT p and NOT d

Now write the conjunction of the two statements: p and d

Jovie is Jovie is notnot a a good dancer and Noel good dancer and Noel is is notnot a superb artist. a superb artist.

Jovie is a good Jovie is a good dancer and Noel is a superb dancer and Noel is a superb artist.artist.

This will be your second statement.

This will be your first statement.

Noel is a superb artist.Noel is a superb artist.

Jovie is a good dancer.Jovie is a good dancer.



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The Connectives - ConjunctionThe Connectives - Conjunction



Consider the following statements:p: Today is Tuesday.q: Tonight is the first track meet.p Λ q : Today is Tuesday and tonight is the first track meet.


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TTTT

TT TT

TTFFFF

FF

FF

FFFFFF

pp qq p Λ q

When given two When given two statements, typically the statements, typically the first statement is TTFF.first statement is TTFF.

The second statement The second statement will alternate TFTF.will alternate TFTF.

A conjunction is true if A conjunction is true if and only if and only if bothboth of its of its statements are true.statements are true.

Since q Since q is is

false…false…

……the the conjunction conjunction

is false.is false.Since p Since p is is

false…false…

……the the conjunction conjunction

is false.is false.


Given the following statements, state the specified conjunctions and identify its truth value.

a: Jupiter is the biggest planet in the solar system.b: A dolphin is just a big fish.c: The sun is not the biggest star in the universe.d: A geometric theorem does not need a proof.e: The Philippines is a member of the United Nations.

a Λ e : T Λ T TRUE

b Λ c : F Λ T FALSE

e Λ c : T Λ T TRUE

d Λ b : F Λ F FALSE

~a Λ e: F Λ T FALSE

e Λ ~d: T Λ T TRUE

~b Λ c: T Λ T TRUE

~b Λ ~d : T Λ T TRUE




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A disjunctiondisjunction conveys the notion of or. We use the symbol V to represent a disjunction.

The ConnectivesThe Connectives

u : Human population will increase.c : Raw resources will be depleted.u V c : Human population will increase or raw resources will be depleted.~u V c : Human population will not increase or raw resources will be depleted.



In everyday life, “or” means one or the other but not both. This is called the exclusive or. In logic, “or” means one or the other or both, called the inclusive or.


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DisjunctionDisjunctionp q p ∨ qT T

T F

F T

F F

A disjunction is A disjunction is falsefalse if and if and only if only if bothboth of its statements of its statements are false.are false.

.bowlinggowillheor

p

swimminggowillJohn q

This statement is This statement is false only if John false only if John does neither.does neither.

TT

TT

TT

FF

He goes swimming….

…so the disjunction

is true.

He goes bowling….

…so the disjunction

is true.

He does BOTH so the disjunction is

TRUE.

He does NEITHER

so the disjunction is FALSE.



p: The biggest continent of Earth is Europe.q: Niel Armstrong is not the first man on the moon.r: All sharks eat people.s: The sum of 3 and 7 is not 9.t: An elephant is the largest land mammal.

s V q : T V F TRUE

a V t : F V T TRUE

q V r : T V F TRUE

r V p : F V F FALSE

~q V r: T V F TRUE

p V ~s: F V F FALSE

~s V ~t: F V F FALSE

~p V ~r : T V T TRUE




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Negation “NOT” ~Negation “NOT” ~Truth value is the opposite of the original statement

Conjunction “AND” Conjunction “AND” ΛΛTruth value of a conjunction is true ONLY if both statements are true

Disjunction “OR” VDisjunction “OR” VTruth value of a disjunction is false ONLY if both statements are false



j: The moon does not revolve around the earth.k: The ostrich is the world’s biggest bird.l: Mt. Carmel is the tallest mountain in the world.m: Euclid is the father of Geometry.n: The Statue of Liberty is not found in Russia.

j Λ k :

l V m :

n Λ j :

l V J :

~k V m:

~l V ~j :

n Λ ~k :

j Λ ~m:



j: The moon does not revolve around the earth.k: The ostrich is the world’s biggest bird.l: Mt. Carmel is the tallest mountain in the world.m: Euclid is the father of Geometry.n: The Statue of Liberty is not found in Russia.

j Λ k : F Λ T FALSE

l V m : F V T TRUE

n Λ j : T Λ F FALSE

l V J : F V F FALSE

~k V m: F V T TRUE

~l V ~j : T V T TRUE

n Λ ~k : T Λ F FALSE

j Λ ~m: F Λ F FALSE


Slides after these are for optional study.



Complete the following truth table to negate a conjunction.


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Truth Tables – The Truth Tables – The Negation of a Negation of a ConjunctionConjunction

p q p ∧ q ~(p ∧ q)

T T

T F

F T

F F

A conjunction A conjunction is only true if is only true if

both both statements statements

are trueare true

A negation A negation yields the yields the

opposite of opposite of the the

previous previous statement.statement.

p q p ∧ q ~(p ∧ q)

T T T F

T F F T

F T F T

F F F T

Click to see the solution.Click to see the solution.




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Complete the following truth table for the disjunctions of two negations.

Truth Tables – Truth Tables – The Disjunction The Disjunction

of Two of Two NegationsNegationsp q ~p ~q ~p V ~ q

T T F F F

T F F T T

F T T F T

F F T T T

p q ~p ~q ~p V ~ q

T T

T F

F T

F F

Click for the solution.




Compare the last columns of the two truth tables.

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Truth Tables – Logically Truth Tables – Logically Equivalent Statements and Equivalent Statements and

DeMorgan’s LawDeMorgan’s Law

p q ~p ~q ~p ∨ ~q

T T F F F

T F F T T

F T T F T

F F T T T

p q p ∧ q ~(p ∧ q)

T T T F

T F F T

F T F T

F F F T

When two statements have the When two statements have the same truth values they are said same truth values they are said to be logically equivalent (to be logically equivalent (≡). . Therefore, Therefore, ~(p ∧ q) ≡ ~p ∨ ~q

This is one of DeMorgan’s This is one of DeMorgan’s Laws.Laws.


You have now completed the instructional portion of

this lesson.

You may proceed to more examples or the practice

assignment.




Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Examples:2.2. Simple vs Compound Simple vs Compound

StatementsStatements3.3. Using the Using the

ConnectivesConnectives4.4. NegationNegation5.5. ConjunctionConjunction6.6. DisjunctionDisjunction7.7. Truth Tables and Truth Tables and

Logically Equivalent Logically Equivalent StatementsStatements

There can never be There can never be surprises in logic. surprises in logic.

Ludwig WittgensteinLudwig Wittgenstein



Practice

How complicated How complicated can truth tables be?can truth tables be?

Practice Logic TablesPractice Logic Tables

Click on address below to practice truth tables

http://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.html


Example 1

The Braves won last night.

Back to main example page

Identify each Identify each sentence as sentence as simple or simple or

compound.compound.

SIMPLESIMPLEPhil played the guitar and Phil played the guitar and sang.sang.

COMPOUNDCOMPOUNDSherri talked on the phone Sherri talked on the phone or played bridge all or played bridge all evening.evening.

COMPOUNDCOMPOUNDDan is not mad at me.Dan is not mad at me.

COMPOUNDCOMPOUND

Only Only one one ideaidea

Two ideas…Two ideas…conjunctionconjunction

Two Two ideas…ideas…

disjunctiondisjunction

NegationNegation


q∨ ∧∼ r

Example 2p


Express each of the Express each of the symbolic statements in symbolic statements in words. words.

pp = “You like to paint,” = “You like to paint,”

qq = “You are an artist,” = “You are an artist,”

rr = “You draw = “You draw landscapes.” landscapes.”

You like to paintYou like to paint

~r

You don’t like to drawYou don’t like to draw landscapeslandscapes

~pYou don’t like to paintYou don’t like to paint

∧ ∼ q

andyou are not an artist.

∨ p

ororyou like to paint.you like to paint.

ororyou are an artistyou are an artist andandyou don’t draw landscapes.you don’t draw landscapes.


Example 3I am not angry at you!


Write the negation of each statement.

I am angry at you!I am angry at you!

My best friend is My best friend is coming over tonight.coming over tonight.

My best friend is not My best friend is not coming over tonight.coming over tonight.

The polygon is a not a The polygon is a not a regular polygon.regular polygon.

The polygon is a The polygon is a regular polygon.regular polygon.

notnot


Example 4George Washington was the first president of the U.S. and John Adams was the second.


Use the truth table to determine whether the following conjunctions are true or false.

True, because both True, because both parts are true.parts are true.

The sum of the measures The sum of the measures of the angles of a pentagon of the angles of a pentagon is 720° and red is a color.is 720° and red is a color.

False. Even though the False. Even though the second part is true, the second part is true, the first part is false. first part is false.

pp qq p p ΛΛ q q

TT TT TT

TT FF FF

FF TT FF

FF FF FF

George George Washington Washington was the first was the first president.president.

John Adams John Adams was the was the second second

president.president.TrueTrue

The sum of The sum of pentagon’s pentagon’s angles is angles is

720720oo.. TrueTrue

TT

FalsFalsee

Red is a Red is a color.color.

TrueTrue

FF


Use the truth table to determine whether the following disjunctions are true or false.

False

True5 - 3 =

2

Dogs can play golf.

Example 5A square is a rectangle or a pentagon has five sides.


True, because both True, because both parts are true. parts are true.

Dogs can play golf or 5 – Dogs can play golf or 5 – 3 = 2.3 = 2.

True. Even though True. Even though the first part is the first part is false, the second false, the second part is true. part is true.

pp qq p p ∨∨ q q

TT TT TT

TT FF TT

FF TT TT

FF FF FF

A square is a

rectangle.

A pentagon has five sides.

True

True

T

T


Example 6Back to main example page

Use truth tables to determine if the statement ~p ∧ ~q is logically equivalent to ~(p ∨ q)

p q p ∨ q ~(p ∨ q)

T T T F

T F T F

F T T F

F F F T

p q ~p ~q ~p ∧ ~q

T T F F F

T F F T F

F T T F F

F F T T T

The two statements The two statements are logically are logically equivalent. Recall:equivalent. Recall:

~(p ∨ q) ≡ ~p ∧ ~q is one of De is one of De Morgan’s Laws.Morgan’s Laws.

1 intro to logic

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