LOGIC

1
LOGIC

• A way of thinking!

2
Puzzle

• At the school the other day, I was chatting to my
  colleagues and noticed a number of things. Judy
  has mousey colored hair and the girl with black
  hair was wearing a green dress. Patty is not
  blonde and April does not have brown hair, Chloe
  was wearing a blue dress. The blonde girl was not
  wearing red and April was not wearing green. I
  can't remember which girl was wearing a yellow
  dress. Can you determine the colors of the girl's
  dresses and their hair?

3
Warm-Up

• Arizona State University is the best college in
  America. After all, more people go to ASU than
  any other college.
• You are in this class so you attend ASU.

4
Arguments

• Valid Argument
• Premises (facts or assumptions)
• Conclusions that support premises
• You are in this class so you attend ASU.
• Fallacy (Invalid Argument)
• Conclusion not supported by premises
• Arizona State University is the best college in
  America. After all, more people go to ASU than
  any other college.

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Arguments Valid or Fallacy

• I used a new pencil on the test last week and got
  an A. The pencil helped me get an A.
• Premise I used a new pencil and got an A
• Conclusion The pencil helped me get an A
• I ate pasta last night and today I played my best
  round of golf. If I continue to eat pasta every
  night I will keep getting better.
• Premise I ate pasta last night and played my
  best round of golf
• Conclusion Eating pasta will keep improving my
  scores

6
Arguments Valid or Fallacy

• 115 degree weather occurs in the summer. Its
  115 degrees today, so it must be summer.
• Premise 115 degree weather occurs in the summer.
  It 115 degrees today.
• Conclusion It must be summer
• Valid
• There is no physical proof that Bigfoot exists,
  thus he does not exist.
• Premise Theres no proof of Bigfoot
• Conclusion Bigfoot does not exist

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Arguments Valid or Fallacy

• Judy has mousey hair. The girl with black hair
  is wearing a green dress. Therefore, Judy is not
  wearing a green dress.
• Premise Judy has mousey hair.
• Premise The girl with black hair is wearing a
  green dress.
• Conclusion Judy is not wearing a green dress.

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The Building Blocks of Arguments

• Propositions
• What are they?
• Claims that can be either TRUE OR FALSE
• Generally in the structure of a sentence
• Has a truth value -- can be assigned to be
  true or false

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The Building Blocks of Arguments

• Proposition?
• We are learning mathematics.
• Proposition (True)
• Phoenix is hot in the summer.
• Proposition (True)
• Lets go to the movies.
• Not a Proposition
• 21-199.
• Proposition (False)

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The Building Blocks of Arguments

• Negation of a Proposition (not, or opposite)
• We are learning mathematics.
• We are not learning mathematics. (False)
• Phoenix is hot in the summer.
• Phoenix is not hot in the summer. (False)
• 21-199.
• 21-19?9. (True)
• Double Negation (not not, double opposite)
• In ordinary language
• The school board does not support unequal
  treatment of students.
• does not supportnegation of support
• unequal treatmentnegation of equal

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Logical Connectors

• And Statements (Conjuctions)
• What are they?
• The joining of two propositions (claims) with and
• Examples
• I am in math class and at ASU
• It is cold outside and it is raining.
• I like to golf and ride bikes.

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Logical Connectors

• Truth Table of a Conjunction (?and)
• Lists all possible combinations of true and false
  statements
• I am in math class and at ASU
• pI am in math class
• qI am at ASU
• Truth Value
• I like to golf and ride bikes.

p q p ? q T T T T F F F T F F F F
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Logical Connectors

• Or Statements (Disjunctions)
• What are they?
• The joining of two propositions (claims) with or
• Two Cases
• Exclusive or (one or the other, but not both)
• Soup or salad
• Inclusive or (either one or both)
• Flood, fire, or robbery Insurance
• Logic Assume Inclusive

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Logical Connectors

• Or Statements (Disjunctions)
• Examples
• I am in math class or I am at ASU
• It is cold outside or it is raining.
• I like to golf or ride bikes.

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Logical Connectors

• Truth Table of a Disjunction (?or)
• I am in math class or I am at ASU
• pI am in math class
• qI am at ASU
• Truth Value
• I like to golf or ride bikes.

p q p ? q T T T T F T F T T F F F
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Logical Connectors

• Online Word Search
• Searching through web-pages, it returns a true or
  a false
• Write the Truth Table for searching math or
  logic and table
• What possible word combinations would be returned?

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Conditionals (Ifthen)

• In mathematics these were the foundations of
  advancements and became theorems
• The Connection of Propositions (Conditional
  propositions)
• If you win the lottery, then I will marry you.
• Hypothesis
• You win the lottery (Often label p)
• Conclusion
• I will marry you. (Often label q)
• Rewrite as If p, then q or p?q

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Conditionals (Ifthen)

• Truth Table
• If you win the lottery, then I will marry you.
• What are the possible outcomes?
• Identify the two propositions
• You win the lottery (T or F)
• I will marry you (T or F)

p q p ? q T T T T F F F T T F F T
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Conditionals (Ifthen)

• Rephrasing and Determining Truth Values
• In groups, rewrite in the form if p then q. Also,
  discuss the truth of the conditional (referring
  to the truth table will help with some)
• Being in this classroom is sufficient for being
  at ASU.
• 225 if 349
• You are in Hawaii whenever you are close to the
  ocean.
• Columbus is the capital of Ohio if Scottsdale is
  the capital of Arizona.

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Converse, Inverse, and Contrapositive

• If Kacie Koch is your professor, then you are
  enrolled in MAT142.
• Consider If you are enrolled in MAT142, then
  Kacie Koch is your professor.
• CONVERSE (if q then p)
• Consider If Kacie Koch is not your professor,
  then you are not enrolled in MAT142
• INVERSE (if not p then not q)
• Consider If you are not enrolled in MAT142, then
  Kacie Koch is not your professor.
• CONTRAPOSITIVE (if not q then not p)

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Converse, Inverse, and Contrapositive

• Truth Values
• If Kacie Koch is your professor, then you are
  enrolled in MAT142.

converse
inverse
contrapositive