Geometry Lesson 9- Introduction to Logic

7/28/2019 Geometry Lesson 9- Introduction to Logic

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Geometry

Lesson 9

Objectives

1. Conditional Statements2. Manipulations of Conditional StatementsVocab

1. 2. 3. 4.Time Activity

Do Now: Examine each of the following real newspaper headlines. When the author

wrote the headline, they had one meaning in mind, but when you read the headline

another funnier meaning should emerge.

(1)Police begin campaign to run down jaywalkers(2)Safety experts say school bus passengers should be belted(3)Panda mating fails; Veterinarian takes over(4)Eye drops off shelf(5)Squad helps dog bite victim(6)Enraged cow injures farmer with ax(7)Miners refuse to work after death(8)Juvenile court to try shooting defendant(9)Stolen painting found by tree(10)Two soviet ships collide, one dies(11)2 sisters reunited after 18 years in checkout counter(12)Drunken drivers paid $1000 in 84

Logic and Conditional Statements: As the do now illustrates, language can be

misleading. One person may intend to say one thing, but the other person hears

something totally different. This leads us to some problems in math:

We have to use words in math. We use them in definitions, in stating rules, inposing problems.

Math has to mean the same thing to everyone!!! If I write down a mathematicalrule, even if I use words, it has to have the same meaning to every person who

reads it or else mathematicians in different locations will each have their own set

of rules and definitions and no one can build on what others have created.

o Math is collaborativemeaning weve all built its rules up together.o Math is also absolute meaning when something is stated mathematically,

it must be true to everyone and everywhere. This brings us to a conclusion: when we use words in math, we must all agree

upon rules those words will follow so that we all understand what the others are

saying.

Exercise 1: Write down three rules you live your life by. They may be rules imposed by

parents or schools or the government, or rules you set for yourself.

(1)Rule 1:


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weve used the correct formatting doesnt mean our statement is now definitely true.

This leads us to the idea of a truth value.

(1)Ex: Let the statement be Lizzy is a teacher. Let the statement be Lizzy eatsstudents for breakfast.

a. What is the truth value of? (T or F)b. What is the truth value of (T or F)c. What is the truth value of (T or F)d. What is the truth value of? (T or F)e. What is the truth value of? (T or F)f. Write the conditional statement .g. What is the truth value of h. Write the negation of the conditional statement, i. What is the truth value of ?

(1)A truth table helps us figure out the different combinations of truth values thatcan exist for a statement. Fill in the truth table for a statement and its negation:

(2)The truth values of conditional statements can get a little tricky. Something thatcan help us understand the truth of conditional statements is called a Euler

Diagram. It shows you physically how a conditional statement works. Consider

the statement if it is snowing, then it is cold

outside.

Notice how in order for it to be snowing, itmust also be cold outsidesince the it is

snowing circle is inside the it is cold outside

circle.

So if you want to draw a Euler Diagram for , is the inside circle and is the outside circle.

Manipulations of Conditional Statements: Now we are ready to examine the truth

value of conditional statements and their manipulations.

Exercise 1:Your parent says to you if you get an A in geometry, then I will buy you a

new graphing calculator.

Let be you get an A in geometryLet be I will buy you a new graphing calculator

Truth Value

A statement has only two states, either true or false.

Negation

You can reverse the state of a conditional statement by adding not to the conclusion.This form of the statement is called the negation of the statement. We represent the

It is

snowing

It is cold

outside


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(1)Case 1: You got an A in geometry and your parentbuys you a calculator. This means the original

promise was a true promise. What was promised was given therefore the

original statement was true. Fill out the truth table for this scenario.

(2)Case 2: You got an A in geometry but your parentrefuses to get you the calculator. The promise wasbroken! The original statement was a big FATLIE! Fill out the truth table for this scenario.

(3)Case 3: You did NOT get an A, but your parent getsyou a graphing calculator anyway in hopes that it

will help you get an A next time. Your parent didnt

break the promise, they just did something nice so

the promise still stands.

(4)Case 4: You did Not get an A so your parent didNOT get you a graphing calculator. The promise

was upheld. Fill in the truth table:

(5)So the complete truth table that maps out all thepossibilities of a conditional statement and tells you

whether or not the conditional statement will be true

in different situations is:

(6)The only time a conditional statement is false iswhen: a true hypothesis does not lead to a true

conclusion.

Exercise 2: Examples. In each of the following, determine if the conditional statement

is true.

(1) If , then

(2) If 2 is a prime number, then 2 is odd.

(3) If 12 is a multiple of 9, then 12 is a multiple of 3.

(4) If then is a positive integer.

T T T

T F F

F T T

F F T

T T T

T F F

F T T

F F T


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Exercise 1: Converse, Inverse and Contrapositive

(1)Write and examine the 4 statements below. is I drink bad milk. is I getsick

a. Write the statement b. What is the truth value of ?c.

Write the statement d. What is the truth value of ?

e. Write the statement f. What is the truth value of?g. Write the statement h. What is the truth value of

Geometry Name:_______________

Lesson 9 Date:_________

Logic and Conditional Statements Class Work

Do Now: Examine each of the following real newspaper headlines. When the author wrote the

headline, they had one meaning in mind, but when you read the headline another funnier meaning

should emerge.

(1)Police begin campaign to run downjaywalkers

(2)Safety experts say school bus passengersshould be belted


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(3)Panda mating fails; Veterinarian takesover

(4)Eye drops off shelf(5)Squad helps dog bite victim

(6)Enraged cow injures farmer with ax(7)Miners refuse to work after death

(8)Juvenile court to try shooting defendant

(9)Stolen painting found by tree(10)Two soviet ships collide, one dies(11)2 sisters reunited after 18 years in

checkout counter

(12)Drunken drivers paid $1000 in 84

Logic and Conditional Statements: As the do now illustrates, language can be misleading.

One person may intend to say one thing, but the other person hears something totally different.

This leads us to some problems in math:

We have to use words in math. We use them in definitions, in stating rules, in posingproblems.

Math has to mean the same thing to everyone!!! If I write down a mathematical rule, even if Iuse words, it has to have the same meaning to every person who reads it or else

mathematicians in different locations will each have their own set of rules and definitions

and no one can build on what others have created.

o Math is collaborativemeaning weve all built its rules up together.o Math is also absolute meaning when something is stated mathematically, it must be

true to everyone and everywhere.

This brings us to a conclusion: when we use words in math, we must all agree upon rulesthose words will follow so that we all understand what others are saying.

Exercise 1: Write down three rules you live your life by. They may be rules imposed by parents or

schools or the government, or rules you set for yourself.

(4)Rule 1:____________________________________________________________________In If-Then form:_____________________________________________________________



Exercise 2: If-Then Statements


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(1)Ex: If its raining meatballs, then the laws of physics have broken.a. What is the hypothesis? b. What is the conclusion?

(2)Ex: Good students get As.a. Rewrite the rule above as a conditional statement

b. What is the hypothesis? c. What is the conclusion?

(3)Ex: Pigs can fly.a. Rewrite the rule above as a conditional statement


(4)Ex: I need bucket-loads of coffee in the morning.a. Rewrite the rule above as a conditional statement


NOTE: Because mathematicians are LAZYthey dont like to write out the whole statement all the

time so they usually choose a letter to represent each statement. A conditional statement is

actually two statements linked by the if-then structure so we let the hypothesis be and theconclusion be . Then we let the arrowstand for if-then.

(5) Let represent the statementyou (will) become president, let represent the statement coffeefor everyone, let represent the statement we all run around frantically.

Conditional Statement

All rules are created out of two statements- one that is preceded with if and one that is

preceded with then. The new sentence thats created is called a conditional statement.

The If statement is called the hypothesis.

It sets a condition that when fulfilled means the then statement called the


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a. Write the statement

b. Write the statement

c. Write the statement

Exercise 3: Notice that the statements above arent necessarily true. Just because weve used the

correct formatting doesnt mean our statement is now definitely true. This leads us to the idea of a

truth value.

(1)Ex: Let the statement be Lizzy is a teacher. Let the statement be Lizzy eats studentsfor breakfast.

a. What is the truth value of? (T or F) b. What is the truth value of (T or F)

c. What is the truth value of d. What is the truth value of()

e. What is the truth value of? f. What is the truth value of?

g. Write the conditional statement . h. What is the truth value of

i. Write the negation of the conditionalstatement,

j. What is the truth value of ?

Truth Value

A statement has only two states, either true or false.

Negation of a statement

You can reverse the state of a statement by adding not. We represent the not in a

negation with the symbol. Two negations cancel each other out. i.e. I am not unhappyimplies I am indeed relatively happy.

Negation of a conditional statement

You can reverse the state of a conditional statementby adding not to the conclusion.


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(2)A truth table helps us figure out the different combinations of truthvalues that can exist for a statement. Fill in the truth table for a

statement and its negation:

(3)The truth values of conditional statements can get a little tricky. Something that can help usunderstand the truth of conditional statements is called a Euler Diagram. It shows you

physically how a conditional statement works. Consider the

statement if it is snowing, then it is cold outside.

Notice how in order for it to be snowing, it must also becold outsidesince the it is snowing circle is inside the it

is cold outside circle.

So if you want to draw a Euler Diagram for , is theinside circle and is the outside circle.

Exercise 4:Now were ready to determine the truth values of conditional statements. Your parent

says to you if you get an A in geometry, then I will buy you a new graphing calculator.

Let be you get an A in geometryLet be I will buy you a new graphing calculator

(1)Case 1: You got an A in geometry and your parent buys you acalculator. This means the original promise was a true promise.

What was promised was given therefore the original statement

was true. Fill out the truth table for this scenario.

(2)Case 2: You got an A in geometry but your parent refuses to getyou the calculator. The promise was broken! The original

statement was a big FAT LIE! Fill out the truth table forthis scenario.

(3)Case 3: You did NOT get an A, but your parent gets you agraphing calculator anyway in hopes that it will help you get an A

next time. Your parent didnt break the promise, they just did

something nice so the promise still stands.

(4)Case 4: You did Not get an A so your parent did NOT get you agraphing calculator. The promise was upheld. Fill in the truth

table:

It is

snowing

It is cold

outside


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(5)So the complete truth table that maps out all the possibilities ofa conditional statement and tells you whether or not the

conditional statement will be true in different situations is:

(6)The only time a conditional statement is false is when:

Exercise 2: Examples. In each of the following, determine if the conditional statement is true.

(1) If , then (2) If 2 is a prime number, then 2 is odd.

(3) If 12 is a multiple of 9, then 12 is amultiple of 3.

(4) If then is a positive integer.

Geometry Name:_______________

Lesson 9 Date:_________

Logic and Conditional Statements Homework

1. Write each of the following as conditional statements, Circle the hypothesis and underlinethe conclusion. Make a Euler diagram to represent the conditional statement and finally

determine if the conditional statement is true or false.

a. If the coffee pot is empty, all the teachers become grumpy.

If-Then form: If the coffee pot is empty, then all the teachers

become grumpy.

True? Y or N (Use truth table to guide you) When the coffee

pot is empty, the teachers are grumpy, so when is true, is

Coffee Pot is

Empty.

Teachers are

Grumpy


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b. It is cold outside if it is snowing

c. Coffee is the drink of the gods

d. If a Tyrannosaurus Rex were alive, He would LOVE hotdogs

e. People who live in glass houses shouldnt throw stones.

f. You will have 7 years of bad luck if you break a mirror.

If-Then form:

True? Y or N (Use truth table to guide you)

If-Then form:


If-Then form:


If-Then form:


If-Then form:


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g. Vampires drink blood.

h. Two points determine a line

2. Ex: Let the statement be Math is Awesome. Let the statement be Everyone lovesmath.

a. What is the truth value of

? (T or F)

Explain.

b. What is the truth value of

(T or F).

Explain.

c. What is the truth value of d. What is the truth value of()

If-Then form:


If-Then form:



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e. What is the truth value of? f. What is the truth value of?

g. Write the conditional statement . h. What is the truth value of

i. Write the negation of the conditionalstatement,

j. What is the truth value of ?

k. Write the conditional statement.

l. What is the truth value of

m. Write the conditional statement.

n. What is the truth value of

3. Try making a truth table that shows in general when the statement is true.


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a. First Fill out all the possible combinations for thetruth values of and

b. Then looking at the truth values of, determinethe truth values of

c. Finally, recalling that a conditional statement isfalse only when the hypothesis

is true but the

conclusion is false, fill out the column furthest to the right.

d. Thinking just about and , under what conditions is the conditional statement false?

4. Lets see if we can make that truth table above make sense.a. Choose a conditional statement that is true (If it is snowing, then it is cold is a good

one. Assume both and are true.) Write the conditional statement . Does thetruth value of this statement match the truth value in the table?

b. Choose a conditional statement where the is true but the is false (If were in NewYork, then it snows in the summer might work.) Write the conditional statement . Does the truth value of this statement match the truth value in the table?

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Geometry Lesson 9- Introduction to Logic