A mathematical sentence is a sentence that states a fact or … · Day 2 - Conjunction, Disjunction, Conditionals, and Biconditionals HW – pages 16-17 #13-34 all, #35 – 65(every

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LOGIC

Name:______________________________

Teacher:____________________________

Pd: _______

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Table of Contents

Day 1 – Introduction to Logic

HW – pages 8-10

Day 2 - Conjunction, Disjunction, Conditionals, and Biconditionals

HW – pages 16-17 #13-34 all, #35 – 65(every other odd)

Day 3 – Truth Tables, Tautologies and Logically Equivalent Statements

HW – pages 21 - 22 #’s 1, 5, 8, 12, 18, 23

Day 4 – Law of Detachment

HW – pages 27-29 #’s 1-4,15, 25- 28 all

Day 5 – Law of Contrapositive, Inverse, Converse; Proofs in Logic

HW – pages 35 - 37 #’s 9, 11, 15, 29, 32

Day 6 – Law of Modus Tollens; Invalid arguments

HW – pages 43 - 44 #s 2, 6, 10, 12-16 even, 21

Pages 44 - 45 #s 2-28, even

Day 7 – The Chain Rule; Law of Disjunctive Inference

HW – pages 50 – 52 #’s 2, 8, 13, 14, 20, 35

pages 53 - 54 #’s 2, 9, 12, 13, 22, 27, 28

Day 8 – De Morgan’s Law and Law of Simplification

HW - pages 59 – 60 #’s 1 – 29 every other odd

pages 61 - 62 #’s 1-7 odd, 15-21 odd, 27-29 odd

Day 9 – Practice with Logic Proofs

HW – pages 63 – 65

Day 10 – Review Exercises

HW – pages 66 – 68

SUMMARY PAGE – Page 69

Exam –

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Introduction to Logic

Logic is the study of reasoning. All reasoning, mathematical or verbal, is

based on how we put sentences together.

A mathematical sentence is a sentence that states a fact or contains a

complete idea. A sentence that can be judged true or false is called a

statement. In the study of logic, each statement is designated by a letter

(p, q, r, etc.) and is assigned a true value (T for true or F for false.)

Two types of mathematical sentences are

a) open sentence – contains a variable and cannot assign a truth

value.

Ex. She is at the park.

Ex. It is fun.

b) closed sentence (statement) – a sentence that can be judged to be

either true or false.

Ex. The degree measure of a right angle is 900.

Ex. There are eight days in a week.

Example/Practice 1:

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Negation Symbol:

The negation of a statement always has the opposite truth value of the original

statement. It is usually formed by adding the word not to the original

statement.

Ex. p : There are 31 days in January. (True)

~p : There are not 31 days in January. (False)

A statement and its negation have opposite truth values.

Example/Practice 2:

Write the negations for each of the following true sentences.

1. Today’s weather is sunny. __________________________________

2. Alice is not going to the play. ________________________________

3. x 2 ________________________________

Example 3:

Answers:

a. ___________________________________ b. ___________

a. ___________________________________ b. ___________

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EXAMPLE 4: Let p represent: January has 31 days

Let q represent: Christmas is in December

For each given sentence:

a. Write the sentence in symbolic form using the symbols below.

b. Tell whether the sentence is true or false.

1. It is not the case that January has 31 days a. _______________ b. _____

2. It is not true that Christmas is not in December a. _______________ b. _____

3. January has 31 days. a. _______________ b. _____

Truth Table

To study sentences where we wish to consider all truth values that could be assigned, we use a

device called a truth table. A truth table is a compact way of listing symbols to show all possible

truth values for a set of sentences.

Example 5: Fill in all missing symbols.

Translating Statements from Words to Symbols

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Example 6: Let p represents “The bird has wings”

Let q represents “The bird can fly”

Write each of the following sentences in symbolic form.

1. The bird has wings and the bird cannot fly.

__________________

2. If the bird can fly, then the bird has wings.

__________________

3. The bird cannot fly or the bird has wings.

__________________

4. The bird can fly if and only if the bird has wings.

__________________

Challenge Problem

Summary/Closure

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Exit Ticket

1.

2.

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DAY 1 – Homework)

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Translating Statements From Words To Symbols

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Conjunctions and Disjunctions

SWBAT:

UWarm - Up

In logic, a ________________________ is a compound sentence formed by combining

two sentences (or facts) using the word _______________

p: Wash the dishes.

q: Vacuum the house.

p ^ q: _________________________________________________________________

For a conjunction (and) to be true, _____________ facts must be true.

q p ^ qp

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In logic, a _________________________ is a compound sentence formed by combining

two sentences (or facts) using the word __________________

p: Wash the dishes.

q: Vacuum the house.

p v q: __________________________________________________________________

For a disjunction (or) to be true, ____________________________ facts must be true.

q p V qp

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UConditional U: if…then

In logic, a ___________________ is a compound statement formed by

combining 2 sentences (or facts) using the words __________________

Refer to the statement: If I come to school then Ms. Williams will give me an A.

Let p: I come to school.

Let q: Ms. Williams gives me an A.

Based on this condition, fill in the accompanying chart.

p q p q

T T

T F

F T

F F

A_________ is a compound sentence formed by combining the

two conditionals: p q and q p.

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Summary of Compound Sentences

A conjunction is a compound sentence formed by using the word

and to combine two simple sentences.

p q

pq

T T

T F

F T

F F

A disjunction is a compound sentence formed by using the word

or to combine two simple sentences.

p q

pq

T T

T F

F T

F F

A conditional is a compound sentence usually formed by using

the words if…then to combine two simple sentences.

p q p q

T T

T F

F T

F F

A biconditional is a compound sentence formed by combining the

two conditionals p q and q p.

p q p q

T T

T F

F T

F F

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Challenge Problem:

Directions: Complete the truth table

Exit Ticket:

p V qqpp q

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HW – Day 2 Assignment

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Truth Tables, Tautologies, and Logically Equivalent Statements

SWBAT:

UWarm - Up

In logic, a tautology is a compound statement that is always true, no matter what truth

values are assigned to the simple sentences within the compound sentence.

Example:

Example:

pq (pq)

(pq)(pq)

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Logically Equivalent Statements

When two statements have the same truth values, we say that the statements are

logically equivalent. To show that equivalence exists between two statements, we use

the biconditional, if and only if: If the result is a tautology then the statements are

logically equivalent.

Ex.1

Let p represent “I study.”

Let q represent “I’ll pass the test.”

The two statements being tested for an equivalence are:

If I study, then I’ll pass the test.

I don’t study or I’ll pass the test.

a) Express each statement in symbolic form.

b) Express the biconditional in symbolic form.

c) Set up a truth table to see if the statements are logically equivalent.

Ex. 2

Let c represent “Simon takes chorus.”

Let s represent “Simon takes Spanish.”

a) Using s and c and proper logic connectives, express each of the following sentences in

symbolic form.

If Simon takes chorus, then he cannot take Spanish.

If Simon takes Spanish then he cannot take chorus.

b) Prove that the two statements are logically equivalent, or give a reason why they are

not equivalent.

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SUMMARY

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HW Day 3 – Tautologies and Logically Equivalent Statements

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Law of Detachment

SWBAT:

UWarm - Up

UDeductive reasoningU is the process of using logic to draw conclusions from given facts,

definitions, and properties.

In deductive reasoning, if the given facts are true and you apply the correct logic,

then the conclusion must be true. The Law of Detachment is one valid form of

deductive reasoning.

An argument consists of a series of statements called premises and a final statement

called a conclusion. We say that the premises lead to the conclusion, or that the

conclusion follows the premises.

For example:

Premise: If I play baseball, then I need a bat.

UPremise: I play baseball.

Conclusion: I need a bat.

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Example 1:

An auto mechanic knows that if a car has a dead battery, the car will not start. A

mechanic begins work on a car and finds the battery is dead. What conclusion will she

make?

Example 2:

If there is lightning, then it is not safe to be out in the open. Marla sees lightning from the

soccer field. What conclusion will she make?

Example 3:

If it is snowing, then the temperature is less than or equal to 32˚F.

The temperature is 20˚F. What conclusion can you make?

UModel Problems

Example 4:

Let r represent “It is raining.”

Let m represent “I have to mow the lawn.”

Given the following premises:

If it is raining, then I don’t have to mow the lawn.

It is raining.

a. Using r, m, and proper logic connectives, express the premises of this argument in

symbolic form.

b. Write a conclusion in symbolic form.

c. Translate the conclusion into words.

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Hidden Conditionals

Sometimes a conditional does not always use the words if …. then or the word implies.

Such a statement is called a hidden conditional.

Ex. A polygon of three sides is called a triangle.

If a polygon has three sides, then the polygon is called a triangle.

Ex. What’s good for Bill is good for me.

If it’s good for Bill then it’s good for me.

Ex. I’ll get a job when I graduate.

If I graduate, then I’ll get a job.

Ex. Drink milk to stay healthy.

If you drink milk, then you will stay healthy.

Example 5:

Write a valid conclusion for the given premises or indicate that no conclusion is possible.

Premises: If adjacent angles are supplementary, then the angles form a linear pair.

Example 6:

Write a valid conclusion for the given premises or indicate that no conclusion is possible.

Premises: Reading will give a person knowledge.

Mrs. Williams is an avid reader.

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SUMMARY

Exit Ticket

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Day 4 – Law of Detachment HW

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Inverse, Converse, Contrapositive

SWBAT:

UWarm - Up

The converse is formed by interchanging the hypothesis and the conclusion.

The inverse is formed by negating the hypothesis and negating the conclusion.

The contrapositive is formed by negating both the hypothesis and conclusion, and then

interchanging the resulting negations.

Ex. Given the conditional : p → q

Find the converse:

Find the inverse:

Find the contrapositive:

Ex. s: It is spring m: The month is May

Conditional (s →m): If it is spring, then the month is May.

Inverse ( ):

Converse ( ):

Contrapositive ( ):

The inverse and converse of a given conditional do not always have the same

truth value as the given conditional.

The conditional and its contrapositive are logically equivalent statements.

The inverse and converse of a given conditional are logically equivalent to each

other.

Complete the truth tables.

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Find the inverse, converse and contrapositive of each of the given statements:

1. t → ~w

2. ~m → p

3. If you use Tickle deodorant, then you will not have body odor.

4. If a man is honest, then he does not steal.

UPractice

1. Given the true statement: “If a person is eligible to vote, then that person is a citizen.” Which

statement must also be true?

(1) Kayla is not a citizen; therefore, she is not eligible to vote.

(2) Juan is a citizen; therefore, he is eligible to vote.

(3) Marie is not eligible to vote; therefore, she is not a citizen.

(4) Morgan has never voted; therefore, he is not a citizen.

2. What is the inverse of the statement “If Julie works hard, then she succeeds”?

(1) If Julie succeeds, then she works hard.

(2) If Julie does not succeed, then she does not work hard.

(3) If Julie works hard, then she does not succeed.

(4) If Julie does not work hard, then she does not succeed.

3. What is the converse of the statement "If a b c2 2 2 , then ABC is a right triangle"?

(1) If ABC is a right triangle, then a b c2 2 2 .

(2) a b c2 2 2 if, and only if, ABC is a right triangle.

(3) If ABC is not a right triangle, then a b c2 2 2 .

(4) If a b c2 2 2 , then ABC is not a right triangle.

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UII. Law of Contrapositive U:

States that when a conditional premise is true, then the contrapositive of the premise if

true. OR – A conditional and its contrapositive are logically equivalent.

U p → q__U or (p → q) (~q → ~p)

~q → ~p

Proofs in Logic

When we are given a series of premises that are true and we apply laws of

reasoning to reach a conclusion that is true, we say that we are proving an argument by

means of a formal proof. We use two columns. The first column consists of

statements and the second column consists of reasons.

Given the premises: If Joanna saves enough money, then she can buy a bike.

Joanna cannot buy a bike.

Prove the conclusion: Joanna did not save enough money.

Statements Reasons

Your Turn! 4. Given the premises: If Al does not study, then he will fail.

Al did not fail.

Prove the conclusion: Al studied.

**** Make sure you set up a formal proof! ****

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Proofs in Logic

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Exit Ticket

1.

2.

3.

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Day 5 - HW

Inverse, Converse, and Contrapositive

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Law of Modus Tollens and Invalid Arguments

SWBAT:

UWarm - Up

In Ruritania, when Juliet is no longer the queen, then her son Prince Charles will take the

throne as the new king. Let us use these facts to present a third law of reasoning.

Let p represent “Charles is the prince” and let q represent “Juliet is the queen”

Premise (p → q): If Charles is still the prince, then Juliet is still the queen.

UPremise (q): Juliet is no longer the queen.

Conclusion (p): Charles is no longer the prince.

What two laws does the Law of Modus Tollens combine?

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Ex. 1. Write a valid conclusion for the given set of premises:

If I am smart, then I like chemistry.

I do not like chemistry.

Ex. 2. Write a valid conclusion: ~r → ~q

Uq_____

Ex. 3 . Write a formal proof:

Ex 4: Write a formal proof

Given: The scale is broken if Sal weighs less than 150 pounds.

The scale is not broken.

Prove: Sal does not weigh less than 150 pounds.

Let b represent: “ The scale is broken”

Let w represent: “ Sal weighs less than 150 pounds.

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Invalid Arguments

At times, we may be confronted with an argument in which all of its premises are true

but these premises do not always lead to a conclusion that is true. The conclusion

could be true or false. Such an argument is called an Uinvalid argument.

Example 1: The First Invalid Argument

Prove why this is invalid.

Example 2:

Prove why this is invalid.

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Model Problems

In 1-4, if the argument is valid, state the law of reasoning that tells why the conclusion

is true. If the argument is invalid, write UInvalid.

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SUMMARY

U

SUMMARY

EXIT TICKET

1.

2.

3.

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Day 6 – Law of Modus Tollens/Invalid Arguments HW

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Invalid Arguments

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The Chain Rule and The Law of Disjunctive Inference

SWBAT:

UWarm - Up

Ex 3: If Phil is not on time, then he will be fired.

If Phil is fired, then Trude will get his job.

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Ex 4: Given: a → c

~a→b

~c

Prove: b

Ex 5: Given: If I walk, then I don’t run

I do not collapse if I don’t run

I collapse

If I do not walk, then I am out of breath.

Prove: I am out of breath

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V. Law of Disjunctive Inference: In the first four laws of inference presented, one or more of the premises was in the form

of a conditional statement p → q. In this next inference law, there are no conditional

statements in the given premises.

As we watch a mystery show on television, we believe that the criminal is either the

butler or the strange neighbor. We then discover a clue that rules out the neighbor. We

can now infer that the butler did it! Let b represent “The butler is the criminal,” and let n

represent “The neighbor is the criminal.” In words and in symbols we say:

Premise (b n): The butler is the criminal or the neighbor is the criminal.

UPremise (n): The neighbor is not the criminal.

Conclusion (b): The butler is the criminal.

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Ex. 2. Write a formal proof:

Given: c → b

a ~b

c

Prove: a

SUMMARY Exit Ticket

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HW – Day 7: Chain Rule and Law of Disjunctive Inference

Chain Rule

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Law of Disjunctive Inference

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Negations, De Morgan’s Laws, and Law of Simplification

SWBAT:

UWarm - Up

VI. Law of Double Negation

~(~p) and p are logically equivalent statements.

VI. DeMorgan’s Law

The next two laws of inference were discovered by an English mathematician named De

Morgan and they bear his name. These inference rules tell how to negate a conjunction

and how to negate a disjunction.

1. Negation of a Conjunction

The negation of a conjunction of two statements is logically equivalent to

the disjunction of the negation of each.

2. Negation of a Disjunction

The negation of a disjunction of two statements is logically equivalent to

the conjunction of the negation of each.

Ex. Write the statement that is logically equivalent to the given statement:

a) ~(h ~d)

b) It is not true that I will take Gloria to the dance or I will take Patricia

to the dance.

c) What is the negation of ~ t m?

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Ex 2: Write a statement that is logically equivalent to:

It is not true that I will take Gloria to the dance or I will take Patricia to the dance.

Ex 3:

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VIII. The Law of Simplification: states that when a single conjunctive premise is true, it

follows that each of the individual conjuncts must be true.

p

qp

q

qp

IX. The Law of Conjunction: states that when two given premises are true, it follows

that the conjunction of these premises is true.

qp

q

p

X. The Law of Disjunctive Addition: states that when a single premise is true, it

follows that any disjunction that has this premise as one of its disjuncts must also be true.

qp

p

Ex 4: Given: r→ t

mr

t → k

Prove: k

Ex 5:

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SUMMARY

Exit Ticket

1.

2. If the statements c d and p d are true, which statement must

also be true?

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HW – Day 8: De Morgan’s Law, and Law of Simplification

De Morgan’s Law

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Law of Simplification

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Practice with Logic Proofs

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Review Exercises

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SUMMARY PAGE

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HOMEWORK ANSWERS