Week

1
Weeks Schedule

• 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

2
Monday' Schedule

• Warm-ups
• Quiz
• Logic lesson
• Logic assignment

3
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?

4
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?

5
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.

6
ExamplesInductive 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.

7
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 childrens 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?

8
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?

9
Tuesday

• Warm-ups
• Correct Assignment 1.1 logic
• Lesson 1.2 patterns
• Assignment

10
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?

11
Find the pattern and then predict the next image.
12
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)

13
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?

14
Wednesday

• Warm-ups
• Correct 1.2Patterns
• Lesson 1.3 Conditional Statements
• Assignment 1.3Conditional Statements

15
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 Wednesday, then tomorrow is Thursday.

16
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.
17
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, cant, will not, nor,
  wouldnt

Today is Tuesday.
Today is not Tuesday.
There exists a dog that is not brown
All dogs are brown.
18
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.
19
The Inverse Mohawk
20
Contrapositive

• The contrapositive is formed by switching the
  order and making both negative.

If today is Wednesday, then tomorrow is Thursday.
then tomorrow is not Thursday.
If today is not Wednesday,
If tomorrow is not Thursday,
then today is not Wednesday.
21
HINT  Remember that the contrapositive (a big
long word) is really the combining together of
the strategies of two other words  converse and
inverse.
22
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

23
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.

24
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

25
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

26
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

27
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.

28
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.

29
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.

30
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.

31
Thursday

• Warm-ups
• Correct lesson 1.3conditional statements
• Continue lesson 1.3conditional statement written
  in symbolic form
• Assignment 1.3

32
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 ifthen, we use an arrow,
  ?.
• To represent the phrase if and only if, we use a
  two headed arrow, .

33
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.

34
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

35
Symbolic Variations

• Converse
• q ? p
• Inverse
• p ? q
• Contrapositive
• q ? p
• Biconditional
• p q

36
Use the statements to construct the propositions.

• p It is a snake.
• q It has scales.
• 1)
• 2)
• 3)

37
Use the statements to construct the propositions.

• p It is a snake.
• q It has scales.
• 1)
• If it is a snake, then it has scales.
• 2)
• 3)

38
Use the statements to construct the propositions.

• p It is a snake.
• q It has scales.
• 1)
• If it is a snake, then it has scales.
• 2)
• It is not a snake
• 3)

39
Use the statements to construct the propositions.

• p It is a snake.
• q It has scales.
• 1)
• If it is a snake, then it has scales.
• 2)
• It is not a snake
• 3)
• If it is not a snake, then it does not have
  scales.

40
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.

41
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!

42
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.

43
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.

44
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 Fridays game.
•  
• 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.
• 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.

Law of Detachment
Law of Syllogism
Not a valid argument
45
Counterexamples

• To find a counterexample, use the following
  method
• Assume that the hypothesis is TRUE.
• Find any example that would make the conclusion
  FALSE.

46
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.

47
Friday

• Warm-ups
• Correct assignment 1.3 Symbolic Notation
• Lesson 1.4 truth tables
• Assignment

48
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?

49
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.

50
Symbols

• Review symbols
• negation
• if-then (conditional)
• iff (biconditional)
• New symbols
• and
• or

51
RulesLet p and q be propositions

• p and q, denoted by , is true when
  both p and q are true and is false otherwise.

52
Rules Let p and q be propositions

• p and q, denoted by , is true when
  both p and q are true and is false otherwise.
• p or q, denoted by , is false when p
  and q are both false and true otherwise.

53
Rules Let p and q be propositions

• p and q, denoted by , is true when
  both p and q are true and is false otherwise.
• p or q, denoted by , is false when p
  and q are both false and true otherwise.
• If p, then q, denoted by , is
  false when p is true and q is false and true
  otherwise.

54
Rules Let p and q be propositions

• p and q, denoted by , is true when
  both p and q are true and is false otherwise.
• p or q, denoted by , is false when p
  and q are both false and true otherwise.
• If p, then q, denoted by , is
  false when p is true and q is false and true
  otherwise.
• p if and only if q, denoted by , is
  true when p and q have the same truth values and
  is false otherwise.

55
Practice
Truth Table for
p q




T
T
T
T
F
F
F
T
F
F
F
F
56
Practice
Truth Table for
p q




T
T
T
T
F
F
F
T
T
F
F
T
57
Practice
Truth Table for
p q




T
T
T
T
F
T
F
T
T
F
F
F
58
Practice
Truth Table for
p q




T
T
T
T
F
F
F
T
F
F
F
T
59
Practice
Truth Table for Table for
p q




T
T
F
F
T
F
T
T
F
T
F
F
F
T
F
F
60
Practice
Truth Table for Table for
p q




T
T
T
T
T
T
F
T
F
F
F
T
T
F
F
F
F
F
F
T
61
Practice
Truth Table for Truth Table for Truth Table for
p q




T
T
T
F
T
T
T
T
F
F
F
T
T
T
F
T
T
T
F
F
T
F
T
F
62
Monday

• Warm-ups
• Correct 1.4 truth tables
• Lesson 1.5 Logical vs Statistical arguments
  Necessary and Sufficient
• Assignment 1.5

63
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.

64
Logical vs Statistical

• Logical Based on reason or what is expected.
• Statistical Based on data, examples,
  experimentation, numerical facts, etc.

65
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.

66
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.

67
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.

68
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.

69
Necessary or Sufficient?

• Earning a total of 95 in class and getting a
  grade of an A.
• Having gas in my car and my car to starting.
• Pouring a gallon of freezing water on my sister
  and her waking up.

Sufficient
Necessary
Sufficient
70
Tuesday
71
Lets 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?

72
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.

73
Algebraic proofs

• An algebraic proof basically involves writing a
  reason for each step while solving an equation.

74
Algebraic properties of equality
Property Definition Identification Abbreviation
Addition Property If ab, then ac bc. Something is added to both sides of the equation. APOE
Subtraction Property If ab, then a-c b-c. Something is subtracted from both sides of the equation. SPOE
Multiplication Property If ab, then ac bc. Something is multiplied to both sides of the equation. MPOE
Division Property If ab and c?0, then a/c b/c. Something is being divided into both sides. DPOE
Property Substitution If ab, 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(bc) ab ac A number outside of parentheses has been multiplied to all numbers inside. DIST
75
Example
Solve 9x1872
Short for Information given to us.
Given
9x1872
SPOE
9x54
x6
DPOE
76
Example
If 5x3x-979, then x11
Given
5x3x-979
8x-979
Dist
x6
DPOE