Unit 1

Logic&

Reasoning

Lesson 1.1

Logic Puzzles

Lesson 1.1 Objectives Utilize deductive reasoning to solve

logic puzzles. (L4.1.1)

Differentiate between inductive v deductive reasoning.

Define Geometry.

Geometry is… Geometry is a branch of mathematics

that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids; the study of properties of given elements that remain invariant under specified transformations.

Basically what that means is geometry is the study of the laws that govern the patterns and elements of mathematics.

Definition from Merriam-Webster Online Dictionary.

Example 1.1 An explorer wishes to cross a barren

desert that requires 6 days to cross. He can only carry his equipment and clothing and nothing else. If one man can only carry enough food for 4 days, what is the fewest number of men traveling on this exploration? 3

Explorer and 2 men carrying food.

Deductive Reasoning Deductive reasoning uses facts, definitions,

and accepted properties in a logical order to write a logical argument.

So deductive reasoning either states laws and/or conditional statements that can be written in if…then form.

There are two laws that govern deductive reasoning.

If the logical argument follows one of those laws, then it is said to be valid, or true.

Example 1.2 A farmer has a fox, goose and a bag of grain, and

one boat to cross a stream, which is only big enough to take one of the three across with him at a time. If left alone together, the fox would eat the goose and the goose would eat the grain. How can the farmer get all three across the stream?

Take OverLeave Behind

Take Back Leave New

Trip 1 Goose Fox & Grain ----- Goose

Trip 2 Fox Grain Goose Fox

Trip 3 Grain Goose ----- Fox & Grain

Trip 4 Goose ----- ----- All

Example 1.3 Sam, Maria, Tim, and Julie are all skilled

at the video game Alien Invaders. Julie scores consistently lower than Tim. Sam is better than Maria, but Maria is better than Tim. Who is the better player, Julie or Maria?

List/Rank1. Sam2. Maria3. Tim4. Julie

Example 1.4 Robert is shopping in a large department store with many floors. He

enters the store on the middle floor from a skyway, and immediately goes to the credit department. After making sure his credit is good, he goes up three floors to the housewares department. Then he goes down five floors to the children’s department. Then he goes up six floors to the TV department. Finally, he goes down ten floors to the main entrance of the store, which is on the first floor, and leaves to go to another store down the street. How many floors does the department store have?

Deductive v Inductive Reasoning Deductive reasoning

uses facts, definitions, and accepted properties in a logical order to write a proof.

This is often called a logical argument.

Inductive reasoning uses patterns of a sample population to predict the behavior of the entire population

This involves making conjectures based on observations of the sample population to describe the entire population.

Examples of Deductive andInductive Reasoning Andrea knows that Robin

is a sophomore and Todd is a junior. All the other juniors that Andrea knows are older than Robin. Therefore, Andrea reasons inductively that Todd is older than Robin based on past observations.

Andrea knows that Todd is older than Chan. She also knows that Chan is older than Robin. Andrea reasons deductively that Todd is older that Robin based on accepted statements.

Homework 1.1 Lesson 1.1 – Logic

p1-2 Due Tomorrow

Lesson 1.2

Patterns

Lesson 1.2 Objectives Describe and predict patterns found in

sequences, figures, and word problems. (L4.1.1)

Define inductive reasoning. Define conjecture. Identify counterexamples for various

conjectures.

Inductive Reasoning Inductive Reasoning is the process in

which one looks for patterns in samples and makes conjectures of how the pattern will work for the entire population.

A conjecture is an unproven statement based on observations.

A conjecture is math’s version of a hypothesis, or educated guess. The education comes from the observation.

Using Inductive Reasoning Much of the reasoning in

Geometry consists of three stages1. Look for a Pattern. Look at examples

and organize any ideas of a pattern into a diagram or table.

2. Make a Conjecture. Use the examples to try to identify what step was taken to get from element to element in the pattern.

3. Verify the Conjecture. Use logical reasoning to verify the conjecture is true for all cases.

Example 1.5

A man starts a chain letter. He sends the letter to two people and asks each of them to send copies to two additional people. These recipients in turn are asked to send copies to two additional people each. Assuming no duplication, how many people will have received copies of the letter after the twentieth mailing? What pattern was being formed with the mailings?

Example 1.6Find the pattern and predict the next figure.1.

2.

3.

Example 1.7Find the pattern and predict the next number.1. 1, 4, 16, 64,…

1. Multiply by 41. 256

2. -5, -2, 4, 13,…2. +3, +6, +9, +12

2. 25

3. 1, 1, 2, 3, 5, 8,…3. Add the previous two numbers in the list

3. 13

4. 1, 2, 4, 7, 11, 16, 22,…4. +1, +2, +3, + 4, +5, +6, +7

4. 29

Example 1.8In order to keep the spectators out of the line of flight, the Air Force arranged the seats for an air show in a “V” shape. Kevin, who loves airplanes, arrived very early and was given the front seat. There were three seats in the second row, and those were filled very quickly. The third row had five seats, which were given to the next five people who came. The following row had seven seats; in fact, this pattern continued all the way back, each row having two more seats than the previous row. The first twenty rows were filled. How many people attended the air show?

Homework 1.2 Lesson 1.2 – Patterns

p3-4 Due Tomorrow

Lesson 1.3

Day 1:Conditional Statements

Lesson 1.3 Objectives Write conditional statements. (L4.2.1) Write the inverse, converse, and contrapositive of

a conditional statement. (L4.2.2) Create a negation of a statement, including

“there exists” and “all” statements. (L4.2.3) Write a biconditional statement. (L4.2.4) Utilize symbolic form of if-then, inverse,

converse, contrapositive, negation, and biconditionals. (L4.3.1)

Apply the laws of detachment and syllogism. (L4.3.2)

Identify a counterexample.

Conditional Statements A conditional statement is any

statement that is written, or can be written, in the if-then form. This is a logical statement that

contains two parts Hypothesis Conclusion

If today is Tuesday, then tomorrow is Wednesday.

Hypothesis The hypothesis of a conditional

statement is the portion that has, or can be written, with the word if in front. When asked to identify the

hypothesis, you do not include the word if.

If today is Tuesday, then tomorrow is Wednesday.

Conclusion The conclusion of a conditional

statement is the portion that has, or can be written with, the phrase then in front of it. Again, do not include the word then

when asked to identify the conclusion.

If today is Tuesday, then tomorrow is Wednesday.

Example 1.9

Write the statements in if-then form.1. Today is Monday. Tomorrow is

Tuesday.1. If today is Monday, then tomorrow is

Tuesday.

2. Today is sunny. It is warm outside.2. If today is sunny, then it is warm outside.

3. It is snowing outside. It is cold.3. If it is snowing outside, then it is cold.

Converse The converse of a conditional

statement is formed by switching the hypothesis and conclusion.

If tomorrow is Wednesday,

If today is Tuesday, then tomorrow is Wednesday.

then today is Tuesday

Negation The negation is the opposite of

the original statement. Make the statement negative of what

it was. Use phrases like

Not, no, un, never, can’t, will not, nor, wouldn’t, etc.

Today is Tuesday. Today is not Tuesday.

Example 1.10

Write the negation of the following statements.

1. It is sunny outside.1. It is not sunny outside.

2. I am not happy.2. I am happy.

3. My dog is black.3. My dog is not black.

Inverse The inverse is found by negating

the hypothesis and the conclusion. Notice the order remains the same!

If today is not Tuesday,

If today is Tuesday, then tomorrow is Wednesday.

then tomorrow is not Wednesday.

Contrapositive The contrapositive is formed by

switching the order and making both negative.

If tomorrow is not Wednesday,

If today is Tuesday, then tomorrow is Wednesday.

If today is not Tuesday, then tomorrow is not Wednesday.

then today is not Tuesday.

Example 1.11Write the converse , inverse , and contrapositive of

the following statements.1. If you get a 60% in the class, then you will pass.

1. Converse – If you pass the class, then you get a 60%.Inverse – If you do not get a 60% in the class, then you will not pass.Contrapositive – If you do not pass the class, then you did not get a 60%.

If there is snow on the ground, then the flowers are not in bloom.

2. Converse – If the flowers are not in bloom, then there is snow on the ground.Inverse – If there is no snow on the ground, then the flowers are in bloom.Contrapositive – If the flowers are in bloom, then there is no snow on the ground.

Biconditional Statement A biconditional statement is a

statement that is written, or can be written, with the phrase if and only if. If and only if can be written shorthand by iff.

Writing a biconditional is equivalent to writing a conditional and its converse.

All definitions are biconditional statements.

Example 1.12Write the conditional statement as a biconditional

statement .1. If the ceiling fan runs, then the light switch is

on.1. The ceiling fan runs if and only if the light switch is on.

2. If you scored a touchdown, then the ball crossed the goal line.

2. You scored a touchdown if and only if the ball crossed the goal line.

3. If the heat is on, then it is cold outside.3. The heat is on if and only if it is cold outside.

Equivalent StatementsCondition

alConverse Inverse Contrapositi

veIf p, then q If q, then p If ~p, then ~q If ~q, then ~p

Written just as it shows in the problem.

Switch the hypothesis

with the conclusion.

Take the original

conditional statement and

make both parts

negative.

Take the converse and

make both parts negative.

If the conditional statement is true, then the contrapositive is also true. Therefore they are equivalent statements!

If the converse is true, then the inverse is also true. Therefore they are equivalent statements!

Extreme Negation For extreme negation , such as:

All, Everyone, Nothing, Nobody, etc It is sufficient enough to show that at least one

item can negate the statement.

Phrase Negation

All There exists one that does not…

Everyone There exists one that does not…

Nothing There exist one that…

Nobody There exists one that…

Somebody All… (or Everybody…)

Example 1.13Write the negation for the following

statements.1. All dogs are black.

1. There exists one dog that is not black.

2. Nobody likes tomatoes.2. There exists one person who likes tomatoes.

3. Somebody is going to get in trouble for toilet papering the school.

3. Everybody is going to get in trouble for toilet papering the school.

Lesson 1.3A Homework Lesson 1.3 Day 1 – Conditional

Statements p5-6

Due Tomorrow Quiz Friday

Lessons 1.1-1.3

Lesson 1.3

Day 2:Symbolic Notation

Symbolic Conditional Statements To represent the hypothesis symbolically,

we use the letter p. We are applying algebra to logic by

representing entire phrases using the letter p. To represent the conclusion, we use the

letter q. To represent the phrase if…then, we use

an arrow, . To represent the phrase if and only if, we

use a two headed arrow, .

Example of Symbolic Representation

If today is Tuesday, then tomorrow is Wednesday. p: Today is Tuesday q: Tomorrow is Wednesday

Symbolic form p q

We read it to say “If p then q.”

Negation Recall that negation makes the

statement “negative.” That is done by inserting the words

not, nor, or, neither, etc. The symbol is much like a negative

sign but slightly altered… ~

Symbolic Variations Converse

q p Inverse

~p ~q Contrapositive

~q ~p Biconditional

p q

Example 1.14Use the statements to construct the propositions.

p: It stays warm for a week.q: The apple trees will bloom.

1. p q1. If it stays warm for a week, then the apple trees will bloom.

2. ~ p2. It does not stay warm for a week.

3. ~ p ~ q3. If it does not stay warm for a week, then the apple trees will not

bloom.

4. ~ q ~ p4. If the apple trees will not bloom, then it does not stay warm for a

week.

5. q p5. If the apple trees bloom, then it stays warm for a week.

6. p q6. It stays warm if and only if the apple trees bloom.

Law of Detachment If pq is a true conditional statement

and p is true, then q is true. It should be stated to you that pq is true. Then it will describe that p happened. So you can assume that q is going to

happen also. This law is best recognized when you

are told that the hypothesis of the conditional statement happened first.

Law of Syllogism If pq and qr are true conditional

statements, then pr is true. This is like combining two conditional

statements into one conditional statement. The new conditional statement is found by taking

the hypothesis of the first conditional and using the conclusion of the second.

This law is best recognized when multiple conditional statements are given to you and they share alike phrases.

Example 1.15Are the following arguments valid?

If so, do they use the Law of Detachment or Law of Syllogism?1. Scott knows that if he misses football practice the day

before the game, then he will not be a starting player in the game. Scott misses practice on Thursday so he concludes that he will not be able to start in Friday’s game.

1. Valid - Law of Detachment2. If it is Friday, then I am going to the movies. If I go to the

movies, then I will get popcorn. Since today is Friday, then I will get popcorn.

2. Valid – Law of Syllogism3. If it is Thanksgiving, then I will eat too much. If I eat too

much, then I will get sick. I got sick so it must be Thanksgiving.

3. Invalid – Argument is out of order to use Law of Syllogism

Counterexamples A counterexample is one example

that shows a conjecture is false. Therefore to prove a conjecture is

true, it must be true for all cases.

Conjecture: Every month has at least 30 days.

Counterexample: February has 28 (or 29).

Finding Counterexamples To find a counterexample, use the

following method Assume that the hypothesis is TRUE. Find any example that would make the

conclusion FALSE. For a biconditional statement, you must

prove that both the original conditional statement has no counterexamples and that its converse has no counterexamples. If either of them have a counterexample, then

the whole thing is FALSE.

Example 1.16Find a counterexample for the following

statements.1. If it is a bird, then it can fly.

1. Ostrich, Penguin, Emu, Cassowary, Rhea, Kiwi, and the Inaccessible Island Rail

2. If it can be driven, then it has four wheels.

2. Motorcycle, Three-wheeled ATV, Semi, Dually Pick-up Truck

3. All boats float. 3. Submarine, Titanic

Lesson 1.3B Homework Lesson 1.3 Day 2 – Symbolic Notation

p7-8 Due Tomorrow Quiz Tomorrow

Lesson 1.1-1.3

Lesson 1.4

Truth Tables

Lesson 1.4 Objectives Write a truth table of the connectives

and their negations (L4.2.2)

not and or if…then if and only if

What is a Truth Table? A truth table displays the relationships

between truth-values of propositions. Truth tables are especially useful in determining

the truth-values of complex propositions constructed from simpler propositions.

Building a Truth Table? Every truth table is constructed to verify the validity of every possible outcome of

the individual proposition. So, all truth tables should begin construction in a similar fashion:

1. Create two columns for p and q.1. Even if they are not used that way in the proposition.

2. Fill the columns for p and q with every possible combination of outcomes.2. ie. Both true, both false, only one is true.

3. Add extra columns for any negation of p and q.3. These columns should contain truth-values that are opposite of their original columns.

4. Add extra columns for any intermediate propositions that are used in the final proposition.5. The last column should be the final proposition.

“If…Then” Truth Table Recall that “If p, then q.” can be denoted as:

p q If Mr. Lent wins $1,000,000, then he will give you $100,000.

p q p q

An “if…then” statement will be false when p is TRUE and q is FALSE, and will be true for all other cases.

p : Mr. Lent wins $1,000,000.

q : He will give you $100,000.

p q p q

T T

T F

F T

F F

TFT

T

“And” Truth Table An “and” statement is written as “p and q.” and can be denoted as:

p q Mr. Lent wins $1,000,000 and he will give you $100,000.

p q p q

An “and” statement will be true when BOTH p is TRUE and q is TRUE, and will be false for all other cases.

p : Mr. Lent wins $1,000,000.

q : He will give you $100,000.

p q p qT T

T F

F T

F F

TFF

F

“Or” Truth Table An “or” statement is written as “p or q.” and can be denoted as:

p q Mr. Lent wins $1,000,000 or he will give you $100,000.

p q p q

An “or” statement will be false when BOTH p is FALSE and q is FALSE, and will be true for all other cases.

p : Mr. Lent wins $1,000,000.

q : He will give you $100,000.

p q p qT T

T F

F T

F F

TTT

F

Truth Table Involving Negation Remember to add an extra column for the negated proposition.

p ~ q Mr. Lent wins $1,000,000 and he will not give you $100,000.

p q p ~ q

Remember, an “and” statement will be true when BOTH p is TRUE and q is TRUE, and will be false for all other cases.

p : Mr. Lent wins $1,000,000.

q : He will give you $100,000.

FTF

T

p q ~ q p ~ q

T T

T F

F T

F F

FTF

F

p q ~ q p ~ q

“If and Only If” Truth Table Recall a biconditional is written “p if and only if q.” and is denoted as:

p q Mr. Lent wins $1,000,000 if and only if he will give you $100,000

p q p q

A “if and only if” statement will be true when p and q are BOTH TRUE and when p and q are BOTH FALSE, and will be false for all other cases.

p : Mr. Lent wins $1,000,000.

q : He will give you $100,000.

p q p q

T T

T F

F T

F F

TFF

T

Example 1.17Construct a truth table for:

(p q) (p q)

p q (p q) (p q)p q (p q) (p q)

T T

T F

F T

F F

p q p q (p q) (p q)

T T

T F

F T

F F

p q p q p q (p q) (p q)

T T

T F

F T

F F

T

T

T

F

T

F

F

F

T

F

F

T

p : Mr. Lent wins $1,000,000.

q : He will give you $100,000.

“OR” “AND” “IF…THEN”

Q: Remember when is an “if…

then” statement is false?

A: When the FIRST proposition is

TRUEand the SECOND

proposition is FALSE.

Example 1.18Construct a truth table for:

(p q) (~ p q)

p : Mr. Lent wins $1,000,000.

q : He will give you $100,000.

p q (p q) (~ p q)p q (p q) (~ p q)

T T

T F

F T

F F

p q ~p (p q) (~ p q)

T T

T F

F T

F F

FFT

T

TFT

T

“IF…THEN”p q ~p p q (p q) (~ p q)

T T

T F

F T

F F

p q ~p p q ~p q (p q) (~ p q)

T T

T F

F T

F F

“IF…THEN”

TTT

F

Q: Remember when is an “if…

then” statement is false?

A: When the FIRST proposition is

TRUEand the SECOND

proposition is FALSE.

“OR”

TTT

T

Lesson 1.4 Homework Lesson 1.4 – Truth Tables

p9-11 Due Tomorrow

Lesson 1.5

Logical v Statistical Arguments

UsingNecessary or Sufficient

Conditions

Lesson 1.5 Objectives Distinguish between statistical and

logical arguments. (L4.2.2)

Differentiate between a necessary and a sufficient condition for an argument. (L4.3.3)

Opening “Argument” You are playing a game of Old Maid. The game is

played by drawing cards from your opponent’s hand to make a matching pair from your hand. The loser is the person left holding the Old Maid card after all pairs have been made.

Imagine you are left with 5 cards in your hand and one of them is the Old Maid. Describe the chances of your opponent drawing a card from your hand and leaving you with the Old Maid.

Are the odds more in your favor or your opponent’s in terms of having the Old Maid after that one draw?

Statistical v Logical Argument A statistical

argument is a way to come to a conclusion involving the use of data, numbers, odds, probabilities, percentages, etc.

For example: The chances of you

keeping the Old Maid is 80%.

A logical argument a way to come to a conclusion by using other valid statements such as laws, definitions, postulates, and theorems.

This typically does not involve numbers or data.

For example: The chances of you

keeping the Old Maid is more in your favor because you keep more cards than your opponent draws.

Example 1.19Use the given situation to make a statistical and a logical

argument.1. You are rolling a number cube (dice) with the numbers 1-6 on it. What

is the chance of getting an even number versus an odd.1. Statistical: ½, 50%, 1:2

Logical: Same as getting an odd because there are the same number of each type.

Drawing from a deck that has 10 black cards and 5 red cards, do you think the next card will be red?

2. Statistical: 1/3, 33%, 1:3

Logical: More likely to get a black card since there is more of them. You flip a coin 10 times and 8 times you get a head. Do you think you

will get a head on the next flip? 3. Statistical: ½,,50%, 1:2

Logical: Same as getting a tails because there are only two possible outcomes with each flip.

Necessary Condition A necessary condition of a statement must

be satisfied for the statement itself to be true. For example:

Having gasoline in my car is a necessary condition for my car to start.

If we say “x is a necessary condition for y,” we mean if we don’t have x, then we won’t have y.

Or put differently, without x, you won’t have y. This means that x must happen in order for y to

happen, but it does not mean that having x guarantees that y will happen.

Example: There is gas in the car but the battery is dead.

Sufficient Condition A sufficient condition of a statement is one

that, if satisfied, will make the statement true. For example:

Rain pouring from the sky is a sufficient condition for the ground to be wet.

If we say “x is a sufficient condition for y,” then we mean if we have x, then we know y must follow.

In other words, if we have x we can guarantee we have y.

Example: If it is raining, then we can guarantee that the ground will be wet.

Necessary v Sufficient Remember, necessary

conditions are must haves. So you have to think, can

the conclusion happen without the condition?

If it the conclusion cannot happen, then it must be a necessary condition.

If the conclusion can happen without the condition, then it must not be necessary!

And to recap, a sufficient condition is a condition that guarantees the conclusion.

The conclusion may happen without it, but…

IF the condition occurs, the conclusion MUST happen.

It is a way to make the outcome happen, but it is not the only way.It is possible for a

condition to be necessary and

sufficient.Example: Getting credit for Geometry is a necessary and

sufficient condition for graduation!

Example 1.20Decide the best statement to complete the sentence.1. Having oxygen in the earth's atmosphere is a

(necessary/sufficient) condition for human life.2. Earning a total of 95% in this class is a

(necessary/sufficient) condition for earning a final grade of A.

3. Pouring a gallon of freezing water on my sleeping sister is a (necessary/sufficient) condition to wake her up.

4. Being at least 16 years of age is a (necessary/sufficient) condition for being able to obtain a driver’s license in Michigan.

Lesson 1.5 Homework Lesson 1.5 – Logical/Statistical,

Necessary/Sufficient p12-13

Due Tomorrow

Lesson 1.6

Introduction toProofs

Lesson 1.6 Objectives Create the basic structure for a

proof. (L4.3.1)

Deliver the opening arguments of a proof by contradiction. (L4.3.2)

Review with Algebra Is the following true or false?

5 = 5 True

What about if we added 3 to both sides? 3 + 5 = 3 + 5

True8 = 8

What if we now subtracted 6 from both sides? 8 – 6 = 8 – 6

True2 = 2

What if we now multiply both sides by 8? 8 2 = 8 2

True16 = 16

And now if we divide both sides by 4? 16 4 = 16 4

True4 = 4

What do you observe happened throughout all this manipulation? As long as we performed the same operation on BOTH sides of the equal sign we created

another true, or equivalent, statement.

What is a Proof? A mathematical proof is a sequence of justified

conclusions used to prove the validity of a statement, or conjecture.

It is more than just showing your work! You must state a reason why each step was done. The reasons why are typically found by stating a…

Definition Accepted Property Another Theorem, or Postulate

A mathematical proof shows that a conclusion is true for ALL cases.

These are used to create theorems, which are true statements created as a result of other true statements.

That is the proof process!

Definition of a Postulate A postulate is a rule that is

accepted without a proof. They may also be called an axiom.

Postulates are used together to prove other rules that we call theorems.

Algebraic Proof An algebraic proof involves

solving Algebra equations by providing a reason for each step along the way. Again, it is more than just showing your

work! You must now state a law or property

of Algebra to show why each step was done.

Algebraic Properties of Equality

Property Definition Example Helpful HintAbbreviatio

nAddition Property If a = b,

then a + c = b + c.7 13x When you add the same

number to both sides during solving.

APOE

Subtraction Property If a = b,then a - c = b - c.

8 17x When you subtract the same number to both sides during solving.

SPOE

77

88

Multiplication PropertyIf a = b,then a c = b c. 3

9

x When you multiply the

same number to both sides during solving.

MPOE9 9

Division PropertyIf a = b,then a c = b c.

6 42x When you divide the same number to both sides during solving.

DPOE6 6

Distributive Property

a(b + c) = ab + ac

4( 7) 4 28x x When you multiply a number next to parentheses by everything inside.

Distribute

Combine Like Terms

ax + bx = (a + b)x

3 8 4 2x x ALL OF THIS WILL BE DONE TO ONE SIDE OF THE EQUATION ONLY!

CLT11 2x

Substitution Property

If a = b, thena + b + c = b + b + c.

When you plug a number in for a variable in an equation.

SUBIf a = 6, then find a + 5.

6 + 5 = 11

Recipe for a ProofProve

If 5x – 18 = 3x + 2, then x = 10.

Statements ReasonsStatements Reasons

1. 5x – 18 = 3x + 2

Always rewrite the

problem first.

And the reason why is

to state the GIVEN problem.

And you should know whento stop because it will be th EXACT statement/conclusionyou are trying to show is true.

Statements Reasons

1. 5x – 18 = 3x + 2

1. Given

Statements Reasons

1. 5x – 18 = 3x + 2

1. Given

2. 2x – 18 = 2

Statements Reasons

1. 5x – 18 = 3x + 2

1. Given

2. 2x – 18 = 2 2. SPOE

Statements Reasons

1. 5x – 18 = 3x + 2

1. Given

2. 2x – 18 = 2 2. SPOE

3. 2x = 20

Statements Reasons

1. 5x – 18 = 3x + 2

1. Given

2. 2x – 18 = 2 2. SPOE

3. 2x = 20 3. APOE

Statements Reasons

1. 5x – 18 = 3x + 2

1. Given

2. 2x – 18 = 2 2. SPOE

3. 2x = 20 3. APOE

4. x = 10

Statements Reasons

1. 5x – 18 = 3x + 2

1. Given

2. 2x – 18 = 2 2. SPOE

3. 2x = 20 3. APOE

4. x = 10 4. DPOE

Example 1.21Prove1. If 5x + 3x – 9 = 79, then x = 11.

Statements ReasonsStatements Reasons

1. 5x + 3x – 9 = 79

Statements Reasons

1. 5x + 3x – 9 = 79 1. Given

Statements Reasons

1. 5x + 3x – 9 = 79 1. Given

2. 8x – 9 = 79

Statements Reasons

1. 5x + 3x – 9 = 79 1. Given

2. 8x – 9 = 79 2. CLT

Statements Reasons

1. 5x + 3x – 9 = 79 1. Given

2. 8x – 9 = 79 2. CLT

3. 8x = 88

Statements Reasons

1. 5x + 3x – 9 = 79 1. Given

2. 8x – 9 = 79 2. CLT

3. 8x = 88 3. APOE (Addition Prop.)

Statements Reasons

1. 5x + 3x – 9 = 79 1. Given

2. 8x – 9 = 79 2. CLT

3. 8x = 88 3. APOE (Addition Prop.)

4. x = 11

Statements Reasons

1. 5x + 3x – 9 = 79 1. Given

2. 8x – 9 = 79 2. CLT

3. 8x = 88 3. APOE (Addition Prop.)

4. x = 11 4. DPOE (Division Prop.)

Proof by Contradiction A proof by contradiction proves the given conjecture

true by attempting to prove the opposite is true. The point is: They both can’t be true at the same time!

The first step in a proof by contradiction is to assume the desired conclusion is not correct.

So rewrite the problem by taking the negation of the conclusion only.

And then try to prove it as we have done before. Example of a Proof by Contradiction

The solution to x + 8 = 17 is not 10.

Statements Reasons

1. The solution to x + 8 = 17 is 10. 1. Given

Statements Reasons

1. The solution to x + 8 = 17 is 10. 1. Given

2. 10 + 8 = 17

Statements Reasons

1. The solution to x + 8 = 17 is 10. 1. Given

2. 10 + 8 = 17 2. SUB

Statements Reasons

1. The solution to x + 8 = 17 is 10. 1. Given

2. 10 + 8 = 17 2. SUB

3. 18 = 17 3. C LT

Statements Reasons

1. The solution to x + 8 = 17 is 10. 1. Given

2. 10 + 8 = 17 2. SUB

3. 18 = 17 (Contradiction) 3. C LT

If this way is wrong, then the original conjecture

must be true!

Example 1.22

Write the first step in constructing a proof by contradiction for the following:

1. The solution to x – 8 = 19 is 27.1. The solution to x – 8 = 19 is not 27.

2. 2x + 5 is an odd number.2. 2x + 5 is not an odd number.

Lesson 1.6 Homework Lesson 1.6 – Proofs

p14-16 Due Tomorrow