How to do a Proof

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How to do a Proof

• Using Uno!

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What does it mean to prove something?

• PROOF (pruf)
• noun
• 1. evidence sufficient to establish a thing as
  true, or to produce belief in its truth.
• 2. anything serving as such evidence What proof
  do you have?
• 3. the act of testing or making trial of
  anything test trial to put a thing to the
  proof.
• 4. the establishment of the truth of anything
  demonstration.
• 5. Law. (in judicial proceedings) evidence
  having probative weight.
• 6. the effect of evidence in convincing the
  mind.
• 7. an arithmetical operation serving to check
  the correctness of a calculation.
• 8. Mathematics, Logic. a sequence of steps,
  statements, or demonstrations that leads to a
  valid conclusion.
• adjective
• 9. able to withstand successful in not being
  overcome proof against temptation.
• 10. impenetrable, impervious, or invulnerable
  proof against outside temperature changes.
• 11. used for testing or proving serving as
  proof.
• 12. of tested or proven strength or quality
  proof armor.
• verb (used with object)
• 13. to test examine for flaws, errors, etc.
  check against a standard or standards.

3
Why do a Proof?

• We will be able to show that ideas in Geometry
  will always be true in any situation.
• We can win an argument!

4
Inductive vs. Deductive

• Inductive Reasoning
• Reasoning from detailed facts to general
  principles.
• Any form of reasoning in which the conclusion,
  though supported by the premises, does not follow
  from them necessarily.
• Deductive Reasoning
• Reasoning from the general to the particular.
• A process of reasoning in which a conclusion
  follows necessarily from the premises presented,
  so that the conclusion cannot be false if the
  premises are true.

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Deductive Reasoning Using Syllogisms

• A syllogism is like the Transitive Property in
  Algebra If a b, and b c, then a c.
  
• If you are accepted to Harvard Medial School,
  then you will become a doctor. If you are a
  doctor, then you will be rich. If you go to
  Harvard Medical School, then you will be rich.
• Angle A is 70 degrees. If an angle has a measure
  less than 90, then it is acute. Angle A is acute.
• If JoAnna trick-or-treats, she will get lots of
  candy. If she get lots of candy, she will eat
  it. If she eats it, she will get cavities. If
  JoAnna trick-or-treats, she will get cavities.

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Finish this Syllogism

• If you live in Manhattan, then you live in New
  York.
• If you live in New York, you live in the United
  States.
• If you live in Manhattan, then you live in the
  United States.

7
Finish this Syllogism

• If Henry studies his Algebra, then he will pass
  his test.
• If Henry passes his test, then he will get good
  grades.
• If Henry studies his Algebra, then he will get
  good grades.

8
Finish this Syllogism

• If a number is a whole number, then it is an
  integer.
• If a number is an integer, then it is a rational
  number.
• If a number is a whole number, then it is a
  rational number.

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Finish this Syllogism

• If I drive over glass, then I will get a flat
  tire.
• If I get a flat tire, then I have to change it.
• If I drive over glass, then I have to change a
  tire.

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Building a 2-Column Proof

• We use deductive reasoning to do proofs.
• Ideas must be laid out step by step using
  postulates or proven theorems to build a
  syllogism.

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Postulates and Theorems

• Postulates are big ideas that are accepted as
  universal truths without proof.
• Theorems are ideas that can be proven using
  deductive logic (through syllogisms).

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2-Column Proof Format

• Write the information Given, and what you are
  trying to Prove.
• Draw a T.
• The first column is for statementsthings that
  MUST be true.
• The second column is for reasonsWHY you know it
  is true.

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Writing an Uno Proof

• The rules of Uno are our postulates.
• Use the first card as the Given.
• Use syllogistic logic to list the order in which
  you would have to play the other cards to finally
  be able to play the Prove card.
• Justify your logic in 2-column format.

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3 postulates of Uno!

1. You can play a card of the same color.
2. You can play a card of the same number.
3. You can play a WILD card at any time in order to
   change the color.

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Sample Proof

• Begin with
• List how to play these cards
• To get to

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You dont have to use every postulate you know in
every proof.
Given
Same Color Change Color Same
Color Same Color
Same Color
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Formal 2-Column Proof

• Given Blue 6 Prove Yellow Reverse
• Statements (What Card to Play) Reasons (I
  can play this card because)
• --------------------------------------------------
  -------------------
• 1. Blue 6 1. Given
• 2. Blue Skip 2. Same Color
• 3. Wild Draw 4 3. Change Color
• 4. Yellow 5 4. Same Color
• 5. Yellow 1 5. Same Color
• 6. Yellow Reverse 6. Same Color

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Given
Prove
Using
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Formal 2-Column Proof

• Given Blue 5 Prove Green 6
• Statements (What Comes Next) Reasons (I can
  play this card because)
• --------------------------------------------------
  --------
• 1. Blue 5 1. Given
• 2. Blue 1 2. Same Color
• 3. Green 1 3. Same Number
• 4. Green 6 4. Same Color

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Given
Prove
Using
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Formal 2-Column Proof

• Given Blue Seven Prove Red Nine
• Statements (What Comes Next) Reasons (I
  can play this card because )
• --------------------------------------------------
  -------------------
• 1. Blue 7 1. Given
• 2. Green 7 2. Same Number
• 3. Green 4 3. Same Color
• 4. Yellow 4 4. Same Number
• 5. Yellow 9 5. Same Color
• 6. Red 9 6. Same Number

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Given
Prove
Using
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Formal 2-Column Proof

• Given Red Reverse Prove Green Nine
• Statements (What Card to Play) Reasons (Why
  I can play the card)
• --------------------------------------------------
  -------------------
• 1. Red Reverse 1. Given
• 2. Red 3 2. Same Color
• 3. Blue 3 3. Same Number
• 4. Wild 4. Change Color
• 5. Green 5 5. Same Color
• 6. Green 9 6. Same Color

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Given
Prove
Using
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Formal 2-Column Proof

• Given Yellow Skip Prove Blue 3
• Statements (What Card to Play) Reasons (I
  can play this card because)
• --------------------------------------------------
  -------------------
• 1. Yellow Skip 1. Given
• 2. Yellow 8 2. Same Color
• 3. Red 8 3. Same Number
• 4. Green 8 4. Same Number
• 5. Blue 8 5. Same Number
• 6. Blue 3 6. Same Color

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Given
Prove
Using
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Formal 2-Column Proof

• Given Yellow 8 Prove Blue 1
• Statements Reasons
• --------------------------------------------------
  -------------------
• 1. Yellow 8 1. Given
• 2. Yellow Skip 2. Same Color
• 3. Green Skip 3. Same Number
• 4. Green Draw 2 4. Same Color
• 5. Red Draw 2 5. Same Number
• 6. Red 5 6. Same Color
• 7. Blue 5 7. Same Number
• 8. Blue 1 8. Same Color