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Week’s Schedule
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•
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Mon: Lesson 1.1 Logic
Tue: Lesson 1.2 Patterns
Wed: Lesson 1.3 Conditional Statements
Thu: Lesson 1.3 (continued) conditional
statements in symbolic form
• Fri: truth tables
Monday' Schedule
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•
•
•
Warm-ups
Quiz
Logic lesson
Logic assignment
Introduction
• 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?
Logic
• Read all the information carefully and
completely.
• Decide what the question is asking.
• Organize the important information.
• Use pictures, tables, grids, etc. to help
solve the problem.
• Think creatively
• Does your answer make sense?
Inductive vs Deductive
reasoning
• Deductive reasoning: Uses facts,
definitions, and accepted properties to
write a logical argument.
• Inductive reasoning: Uses previous
examples and patterns to make a
conjecture.
Examples
Inductive or Deductive?
• Andrea knows that Todd is older than Chan. She
also knows that Chan is older than Robin. Andrea
reasons deductively that Todd is older than Robin
based on accepted statements.
• 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.
Practice
• 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?
Practice
• An explorer wishes to cross a barren desert
that requires 6 days to cross, but one man
can only carry enough food for 4 days.
What is the fewest number of other men
required to help carry enough food for him
to cross?
Tuesday
•
•
•
•
Warm-ups
Correct Assignment 1.1 logic
Lesson 1.2 patterns
Assignment
Think about…
• 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?
Find the pattern and then predict
the next image.
Predict the next number in the
sequence. What is the pattern?
1, 4, 16, 64, . . .
256 (multiplied by 4)
–5, -2, 4, 13, . . .
25 (+3, +6, +9, +12)
1, 1, 2, 3, 5, 8, . . .
13 (add previous two to get the next)
1, 2, 4, 7, 11, 16, 22, . . .
29 (+1, +2, +3, +4, etc)
Brain Buster!
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?
Wednesday
•
•
•
•
Warm-ups
Correct 1.2—Patterns
Lesson 1.3 Conditional Statements
Assignment: 1.3—Conditional Statements
Conditional Statements
• A conditional statement is any statement
that is written, or can be written, in the ifthen form.
– This is a logical statement that contains two
parts:
• Hypothesis
• Conclusion
• If today is Wednesday, then tomorrow is
Thursday.
Converse
• The converse of a conditional statement is
formed by switching the hypothesis and
conclusion.
If today is Wednesday, then tomorrow is Thursday.
If tomorrow is Thursday, then today is Wednesday.
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…
Today is Tuesday.
All dogs are brown.
Today is not Tuesday.
There exists a dog
that is not brown
Inverse
• The inverse is found by negating the
hypothesis and the conclusion.
– Notice the order remains the same!
If today is Wednesday, then tomorrow is Thursday.
If today is not Wednesday, then tomorrow is not Thursday.
The Inverse Mohawk
Contrapositive
• The contrapositive is formed by switching
the order and making both negative.
If today is Wednesday, then tomorrow is Thursday.
If today is not Wednesday, then tomorrow is not Thursday.
If tomorrow is not Thursday, then today is not Wednesday.
HINT: Remember that the contrapositive (a big
long word) is really the combining together of the
strategies of two other words: converse and inverse.
Write the statements in
if-then form.
• 1) Today is Monday. Tomorrow is Tuesday.
If today is Monday, then tomorrow is Tuesday.
• 2) Today is sunny. It is warm outside.
If today is sunny, then it is warm outside.
• 3) It is snowing outside. It is cold.
If is is snowing outside, then it is cold
Write the negation of the
following statements.
• 1) It is sunny outside.
• It is not sunny outside.
• 2) I am not happy.
• I am happy.
• 3) All birds can fly.
There exists a bird that cannot fly.
Write the inverse, converse and
contrapositive of the conditional statement.
• Conditional statement: If you get a 60% in the
class, then you will pass.
• Inverse:
• Converse:
• Contrapositive:
Write the inverse, converse and
contrapositive of the conditional statement.
• Conditional statement: If you get a 60% in the
class, then you will pass.
• Inverse: If you do not get a 60% in class, then
you will not pass.
• Converse:
• Contrapositive:
Write the inverse, converse and
contrapositive of the conditional statement.
• Conditional statement: If you get a 60% in the
class, then you will pass.
• Inverse: If you do not get a 60% in class, then
you will not pass.
• Converse: If you pass, then you got a 60% in
class.
• Contrapositive:
Write the inverse, converse and
contrapositive of the conditional statement.
• Conditional statement: If you get a 60% in the
class, then you will pass.
• Inverse: If you do not get a 60% in class, then
you will not pass.
• Converse: If you pass, then you got a 60% in
class.
• Contrapositive: If you do not pass, then you did
not get a 60% in class.
Equivalent statements
• If the conditional statement is true, then the
contrapositive statement is also true.
Therefore, they are equivalent statements.
• If the inverse statement is true, then the
converse statement is also true. Therefore,
they are equivalent statements.
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.
Write the following conditional statements as
biconditional statements.
• 1) If the ceiling fan runs, then the light switch is
on.
– 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.
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.
The heat is on iff it is cold outside.
Thursday
• Warm-ups
• Correct lesson 1.3—conditional statements
• Continue lesson 1.3—conditional
statement written in symbolic form
• Assignment 1.3
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
Use the statements to construct the
propositions.
•
•
p: It is a snake.
q: It has scales.
1)
pq
2)
p
3)
p q
Use the statements to construct the
propositions.
•
p: It is a snake.
•
q: It has scales.
1)
pq
If it is a snake, then it has scales.
2)
p
3)
p q
Use the statements to construct the
propositions.
•
•
1)
p: It is a snake.
q: It has scales.
pq
If it is a snake, then it has scales.
2)
p
It is not a snake
3)
p q
Use the statements to construct the
propositions.
•
•
1)
p: It is a snake.
q: It has scales.
pq
If it is a snake, then it has scales.
2)
p
It is not a snake
3)
p q
If it is not a snake, then it does not have scales.
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.
Example
• If you get a D- or above in Geometry, then
you will get credit for the class.
• Your final grade is a D.
• Therefore…
– You will get credit for this class!
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
• If tomorrow is Wednesday, then the day
after is Thursday.
• If the day after is Thursday, then there is a
quiz on Thursday.
• Therefore…
– And this gets phrased using another
conditional statement
• If tomorrow is Wednesday, then there is a quiz on
Thursday.
Are the following logical arguments? If so do
they use the law of syllogism or detachment?
• 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. 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. 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. Not a valid argument
Counterexamples
• To find a counterexample, use the
following method:
– Assume that the hypothesis is TRUE.
– Find any example that would make the
conclusion FALSE.
Find a counterexample
• If it can be driven, then it has four wheels.
• All boats float.
• If it is a bird, then it can fly.
Friday
•
•
•
•
Warm-ups
Correct assignment 1.3 Symbolic Notation
Lesson 1.4 truth tables
Assignment
Consider the statement:
• “If today is Friday, then 2 + 3 = 6.”
• Is this statement true if it is Wednesday
• Is this statement ever true or is it always false?
• “If it is sunny today, then we will go to the
beach.”
• When is this statement true and when is this
statement false?
Truth tables
• A truth table displays the relationships
between the truth values of propositions.
Truth tables are especially useful in
determining the truth values of
propositions constructed from simpler
propositions.
Symbols
• Review symbols:
negation
if-then (conditional)
iff (biconditional)


• New symbols:
and
or 

Rules—Let p and q be propositions
• “p and q”, denoted by p  q , is true when both
p and q are true and is false otherwise.
Rules —Let p and q be propositions
• “p and q”, denoted by p  q , is true when both
p and q are true and is false otherwise.
• “p or q”, denoted by p  q , is false when p and q
are both false and true otherwise.
Rules —Let p and q be propositions
• “p and q”, denoted by p  q , is true when both
p and q are true and is false otherwise.
• “p or q”, denoted by p  q , is false when p and q
are both false and true otherwise.
• “If p, then q”, denoted by p  q , is false when
p is true and q is false and true otherwise.
Rules —Let p and q be propositions
• “p and q”, denoted by p  q , is true when both
p and q are true and is false otherwise.
• “p or q”, denoted by p  q , is false when p and q
are both false and true otherwise.
• “If p, then q”, denoted by p  q , is false when
p is true and q is false and true otherwise.
• “p if and only if q”, denoted by p  q , is true
when p and q have the same truth values and is
false otherwise.
Practice
Truth Table for
pq
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
F
Practice
Truth Table for p  q
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
Practice
Truth Table for p  q
p
q
pq
T
T
T
T
F
T
F
T
T
F
F
F
Practice
Truth Table for p  q
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
T
Practice
Truth Table for p 
q
p
q
q
p
T
T
F
F
T
F
T
T
F
T
F
F
F
F
T
F
q
Practice
Truth Table
( p forq )
 ( 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  q)  ( p  q)
Practice
Truth
Table
(p 
q )for
(
p  q)
( p  q)  ( p  q)
p
q
pq
T
T
T
F
T
T
T
F
F
F
T
T
F
T
T
T
T
T
F
F
T
T
F
F
p
pq
Monday
• Warm-ups
• Correct 1.4 truth tables
• Lesson 1.5 Logical vs Statistical arguments
Necessary and Sufficient
• Assignment 1.5
Logical or Statistical? What kind
of argument will you make?
• You are playing a game of Old Maid. In
your current hand of five cards, one is the
Old Maid. Give an argument explaining
the odds that after your opponent draws,
you are still stuck holding the Old Maid.
Logical vs Statistical
• Logical: Based on reason or what is
expected.
• Statistical: Based on data, examples,
experimentation, numerical facts, etc.
Make a statistical and a logical
argument for the given situation.
• 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.
– Statistical: ½ or 50%. (3 even out of 6 total)
– Logical: Chances are even because there are
the same number as evens as there are odds.
Make a statistical and a logical
argument for the given situation.
• Drawing from a deck that has 10 black
cards and 5 red cards, do you think the next
card will be red?
– Statistical: No, the next card only has a 5 out
of 15, or 33%, chance of being red.
– Logical: No, there are a twice as many black
cards as there are red.
Necessary
• If we say that "x is a necessary condition for y,"
we mean that if we don't have x, then we won't
have y
• To say that x is a necessary condition for y does
not mean that x guarantees y.
• Water is necessary for plant life. However, it
water does not guarantee plant life.
Sufficient
• If we say that "x is a sufficient condition
for y," then we mean that if we have x, we
know that y must follow
• In other words, x guarantees y.
• Rain pouring from the sky is a sufficient
condition for the ground to be wet.
Necessary or Sufficient?
• Earning a total of 95% in class and getting
a grade of an A.
Sufficient
• Having gas in my car and my car to
starting.
Necessary
• Pouring a gallon of freezing water on my
sister and her waking up.
Sufficient
Tuesday
Let’s try…
• Start out with the statement: 5 = 5
• Add 3 to both sides of the equal sign. What do
you notice?
• Subtract 3 from both sides of the equal sign.
What do you notice?
• Divide both sides of the equation by 5. What do
you notice?
• Multiply both sides of the equation by 3. What
do you notice?
Vocabulary
• Theorem: A true statement that follows as a result
of other true statements
• Postulate: A rule that is accepted without proof.
• Proof: A sequence of justified conclusions used
to prove the validity of a statement.
• Conjecture: An unproven statement that is based
on observations.
Algebraic proofs
• An algebraic proof basically involves
writing a reason for each step while
solving an equation.
Algebraic properties of equality
Property
Definition
Identification
Abbreviation
Addition
Property
If a=b, then a+c = b+c.
Something is added to both
sides of the equation.
APOE
Subtraction
Property
If a=b, then a-c = b-c.
Something is subtracted from
both sides of the equation.
SPOE
Multiplication
Property
If a=b, then ac = bc.
Something is multiplied to
both sides of the equation.
MPOE
Division
Property
If a=b and c≠0, then
a/c = b/c.
Something is being divided
into both sides.
DPOE
Property
Substitution
If a=b, then a can be
substituted for b in any
expression.
One object is used in place of
another without any
calculations being done.
SUB
Distributive
Property
a(b+c) = ab + ac
A number outside of
parentheses has been
multiplied to all numbers
inside.
DIST
Example
Solve 9x+18=72
9x+18=72
Given
9x=54
SPOE
-18
9
-18
9
x=6
DPOE
Short for “Information
given to us.”
Example
If 5x+3x-9=79, then x=11
5x+3x-9=79
8x-9=79
9
Given
Dist
9
x=6
DPOE