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

LOGIC
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Day 1 – Introduction to Logic
HW – page 4 in this packet!!!
Day 2 - Conjunction, Disjunction, Conditionals, Biconditionals
HW – pages 1- 3
Day 3 – Truth Tables, Tautologies and Logically Equivalent Statements
HW – pages 4 -5 #’s 1, 5, 8, 12, 18, 23
Day 4 – Law of Detachment
HW – pages 6-7 #’s 3,4,15, 25- 28 all
Day 5 – Law of Contrapositive, Inverse, converse; Proof in Logic
HW – pages 8 -9 #’s 9, 11, 15, 29, 32
Day 6 – Law of Modus Tollens; Invalid arguments
HW – pages10-11 #s 2, 6, 10, 12-16 even, 21
pages12-13 #s 2-28, even
Day 7 – The Chain Rule; Law of Disjunctive Inverse
HW – pages 14 – 15 #’s 2, 8, 13, 14, 20, 35
pages 16- 17 #’s 2, 9, 12, 13, 22, 27, 28
Day 8 – Negations and De Morgan’s Law
HW - pages 18 – 19 #’s 1 – 29 odd
Day 9 – Law of Simplification, Conjunctive Addition And Disjuncitve Addition
HW – page 20-21 #’s 1-7 odd, 15-21 odd, 27-29 odd
Day 10 - Practice with Logic Proofs
HW – pages 22 – 26
Exam – 9/24/2010
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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.
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. Basketball 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.
P
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.
Negation of symbols:
=
≠
≤
≥
≥
≤
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Homework
Determine whether the following statements are true or false or
neither.
U
1.
2.
3.
4.
5.
6.
7.
Today is the first day of school.
The sky is green.
It is purple.
A triangle has four sides.
5 is between 2 and 7.
It is fast.
x is greater than 5.
________
________
________
________
________
________
________
All sentences whose truth values can be determined are called
statements. Using the sentences above, determine which are
statements. Place an S next to the statements.
Write the Negation of:
U
U
1.
2.
3.
4.
5.
6.
It is snowing.
I like Math
6+2=9
It is Thursday
I am not going to School.
5 2
_______________________
_______________________
_______________________
_______________________
_______________________
_______________________
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Conjunctions and Disjunctions
SWBAT:
Warm - Up
U
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.
p
q
p ^ q
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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.
p
q
p Vq
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Conditional : if…then
U
U
In logic, a ___________________ is a compound statement formed by
combining _________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
T
T
F
F
q
T
F
T
F
p
q
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
p
q
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
p
q
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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Truth Tables, Tautologies, and Logically Equivalent Statements
SWBAT:
Warm - Up
U
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: p q
( p q)
Example: (p q)
(p q)
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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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Law of Detachment
SWBAT:
Warm - Up
U
Deductive reasoning is the process of using logic to draw conclusions from given facts,
definitions, and properties.
U
U
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.
Premise:
I play baseball.
Conclusion:
I need a bat.
U
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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?
Model Problems
U
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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Inverse, Converse, Contrapositive
SWBAT:
Warm - Up
U
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.
II. Law of Contrapositive:
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
U
U
p → q__ or (p → q)
~q → ~p
U
(~q → ~p)
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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
Practice
U
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.
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3. What is the converse of the statement "If a 2 b 2 c2 , then
(1) If ABC is a right triangle, then a 2 b2 c2 .
(2) a 2 b2 c2 if, and only if, ABC is a right triangle.
(3) If ABC is not a right triangle, then a 2 b2 c2 .
(4) If a 2 b 2 c2 , then ABC is not a right triangle.
4. Given the premises:
Prove the conclusion:
ABC is a right triangle"?
If Al does not study, then he will fail.
Al did not fail.
Al studied.
**** Make sure you set up a formal proof! ****
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Law of Modus Tollens and Invalid Arguments
SWBAT:
Warm - Up
U
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”
U
Premise (p → q):
Premise ( q):
Conclusion ( p):
If Charles is still the prince, then Juliet is still the queen.
Juliet is no longer the queen.
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
q_____
U
Ex. 3 . 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 invalid argument.
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Example 1: The First Invalid Argument
Example 2:
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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 Invalid.
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The Chain Rule and The Law of Disjunctive Inference
U
SWBAT:
Warm - Up
U
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
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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.
Premise ( n):
The neighbor is not the criminal.
Conclusion (b):
The butler is the criminal.
U
Ex. 2. Write a formal proof:
Given: c → b
a ~b
c
Prove: a
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Negations and De Morgan’s Laws
SWBAT:
Warm - Up
U
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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VII.
Law of Double Negation
~(~p) and p are logically equivalent statements.
Ex. 1. What is the negation ~p
q?
Ex. 2. What is the negation of “ Peas are green and apples are not blue.”?
Ex. 3. 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.
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The Laws of Simplification, Conjunction, and Disjunctive Addition
SWBAT:
Warm - Up
U
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
q
p
p
q
q
IX. The Law of Conjunction: states that when two given premises are true, it follows
that the conjunction of these premises is true.
p
q
p
q
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.
p
p
q
Example: Given: r→ t
r m
t→k
Prove: k
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