Transcript p q - Mona Shores Public Schools

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 Over
Leave 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/Rank
1.
2.
3.
4.
Sam
Maria
Tim
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

argument.

facts, definitions, and
accepted properties in a
logical order to write a proof.
This is often called a logical
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 and
Inductive 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 stages
1.
2.
3.
Look for a Pattern. Look at examples and
organize any ideas of a pattern into a diagram or
table.
Make a Conjecture. Use the examples to try
to identify what step was taken to get from
element to element in the pattern.
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 1.6
Find the pattern and predict the next figure.
1.
2.
3.
Example 1.7
Find the pattern and predict the next number.
1. 1, 4, 16, 64,…
1. Multiply by 4
1.
2.
256
-5, -2, 4, 13,…
2. +3, +6, +9, +12
2.
3.
25
1, 1, 2, 3, 5, 8,…
3. Add the previous two numbers in the list
3.
4.
13
1, 2, 4, 7, 11, 16, 22,…
4. +1, +2, +3, + 4, +5, +6, +7
4.
29
Example 1.8
In 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.
Today is sunny. It is warm outside.
2.
2.
If today is sunny, then it is warm outside.
It is snowing outside. It is cold.
3.
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 today is Tuesday, then tomorrow is Wednesday.
If 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.
2.
I am not happy.
2.
3.
It is not sunny outside.
I am happy.
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 Tuesday, then tomorrow is Wednesday.
If today is not Tuesday, then tomorrow is not
Wednesday.
Contrapositive

The contrapositive is formed by
switching the order and making both
negative.
If today is Tuesday, then tomorrow is Wednesday.
If today is not Tuesday, then tomorrow is not
Wednesday.
If tomorrow is not Wednesday, then today is not
Tuesday.
Example 1.11
Write 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 1.12
Write 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.
If you scored a touchdown, then the ball crossed
the goal line.
2.
2.
You scored a touchdown if and only if the ball crossed the
goal line.
If the heat is on, then it is cold outside.
3.
3.
The heat is on if and only if it is cold outside.
If the conditional statement is true, then the contrapositive
is also true. Therefore they are equivalent statements!
Equivalent Statements
Conditional
Converse
Inverse
Contrapositive
If 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 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
All
Everyone
Nothing
Nobody
Somebody
Negation
There exists one that does not…
There exists one that does not…
There exist one that…
There exists one that…
All… (or Everybody…)
Example 1.13
Write the negation for the following statements.
1.
All dogs are black.
1.
There exists one dog that is not black.
Nobody likes tomatoes.
2.
2.
There exists one person who likes tomatoes.
Somebody is going to get in trouble for
toilet papering the school.
3.
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


Inverse


~p  ~q
Contrapositive


qp
~q  ~p
Biconditional

p
q
Example 1.14
Use the statements to construct the propositions.
p: It stays warm for a week.
q: The apple trees will bloom.
1.
pq
If it stays warm for a week, then the apple trees will bloom.
1.
~p
2.
It does not stay warm for a week.
2.
~p~q
3.
If it does not stay warm for a week, then the apple trees will not bloom.
3.
~q~p
4.
If the apple trees will not bloom, then it does not stay warm for a week.
4.
qp
5.
If the apple trees bloom, then it stays warm for a week.
5.
p
6.
6.
q
It stays warm if and only if the apple trees bloom.
Law of Detachment

If pq is a true conditional statement and p
is true, then q is true.




It should be stated to you that pq 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 pq and qr are true conditional
statements, then pr 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 1.15
Are the following arguments valid?
If so, do they use the Law of Detachment or Law of Syllogism?
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.
1.
Valid - Law of Detachment
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.
2.
Valid – Law of Syllogism
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.
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 1.16
Find 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
If it can be driven, then it has four wheels.
2.
2.
Motorcycle, Three-wheeled ATV, Semi, Dually Pick-up Truck
All boats float.
3.
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.
2.
Fill the columns for p and q with every possible combination of outcomes.
2.
3.
5.
ie. Both true, both false, only one is true.
Add extra columns for any negation of p and q.
3.
4.
Even if they are not used that way in the proposition.
These columns should contain truth-values that are opposite of their original
columns.
Add extra columns for any intermediate propositions that are used in the
final proposition.
The last column should be the final proposition.
p : Mr. Lent wins $1,000,000.
q : He will give you $100,000.
“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
T
T
F
F
q
T
F
T
F
pq
T
F
T
T
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.
“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
T
T
F
F
q
T
F
T
F
pq
T
F
F
F
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.
“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
T
T
F
F
q
T
F
T
F
pq
T
T
T
F
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.
Truth Table Involving Negation

Remember to add an extra column for the negated proposition.

p~q

p
T
T
F
F
Mr. Lent wins $1,000,000 and he will not give you $100,000.
p
q
T
F
T
F
q
~ qp
F
T
F
T
 ~pq  ~ q
F
T
F
F
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.
“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
T
T
F
F
q
T
F
T
F
pq
T
F
F
T
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.
Example 1.17
Q: Remember when is
an “if…then” statement
is false?
A: When the FIRST
proposition is TRUE
and the SECOND
proposition is FALSE.
Construct a truth table for:

(p  q)  (p  q)
“OR”
“AND”
“IF…THEN”
p
q
pq
pq
(p  q)  (p  q)
T
T
T
T
T
T
F
T
F
F
F
T
T
F
F
F
F
F
F
T
p : Mr. Lent wins $1,000,000.
q : He will give you $100,000.
Example 1.18
Q: Remember when is
an “if…then” statement
is false?
A: When the FIRST
proposition is TRUE
and the SECOND
proposition is FALSE.
Construct a truth table for:

p
T
T
F
F
q
T
F
T
F
(p  q)  (~ p  q)
“IF…THEN”
F
F
T
pq
T
F
T
T
T
~p
“IF…THEN”
~p
q
T
T
T
F
“OR”
(p  q)  (~ p  q)
T
T
T
T
Lesson 1.4 Homework

Lesson 1.4 – Truth Tables


p9-11
Due Tomorrow
Lesson 1.5
Logical v Statistical Arguments
Using
Necessary 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 1.19
Use the given situation to make a statistical and a logical
argument.
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.
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!
Example: Getting credit for Geometry
is a necessary and sufficient condition
for graduation!

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 1.20
Decide 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 to
Proofs
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


What about if we added 3 to both sides?

3+5=3+5


8–6=8–6

82=82

True
16 = 16
And now if we divide both sides by 4?

16  4 = 16  4


True
2=2
What if we now multiply both sides by 8?


True
8=8
What if we now subtracted 6 from both sides?


True
True
4=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
Addition Property
Definition
If a = b,
then a + c = b + c.
Subtraction Property
If a = b,
then a - c = b - c.
Multiplication Property
If a = b,
then a  c = b  c.
Division Property
If a = b,
then a  c = b  c.
Example
x  7  13
7
7
x  8  17
8
8
x
 3 9
9
6x  42
9
6
6
Helpful Hint
Abbreviation
When you add the same
number to both sides
during solving.
APOE
When you subtract the
same number to both
sides during solving.
SPOE
When you multiply the
same number to both
sides during solving.
MPOE
When you divide the same
number to both sides
during solving.
DPOE
Distributive Property
a(b + c) = ab + ac
4( x  7)  4 x  28
When you multiply a number
next to parentheses by
everything inside.
Distribute
Combine Like Terms
ax + bx = (a + b)x
3x  8x  4  2
11x  2
ALL OF THIS WILL BE DONE
TO ONE SIDE OF THE
EQUATION ONLY!
CLT
Substitution Property
If a = b, then
a + b + c = b + b + c.
If a = 6, then find a + 5.
When you plug a number in
for a variable in an equation.
6 + 5 = 11
SUB
Recipe for a Proof
Prove
If 5x – 18 = 3x + 2, then x = 10.
Statements
Reasons
Always rewrite the
1. 5x – 18 = 3x + 2
1. Given
problem first.
2. 2x – 18 = 2
2. SPOE
3. 2x = 20
3. APOE
4. x = 10
4. DPOE
And you should know when
to stop because it will be th
EXACT statement/conclusion
you are trying to show is true.
And the reason why is
to state the GIVEN problem.
Example 1.21
Prove
1.
If 5x + 3x – 9 = 79, then 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
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.
2.
The solution to x – 8 = 19 is not 27.
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