How to do a Proof

How to do a ProofHow to do a ProofHow to do a ProofHow to do a Proof

Using Uno!Using Uno!

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.

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!

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.

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.

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.


• 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.


• 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.


• 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.

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.

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).

2-Column Proof Format:

• Write the information Given, and what you are trying to Prove.

• Draw a T.• The first column is for statements

—things that MUST be true.• The second column is for reasons

—WHY you know it is true.

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.

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.

3 postulates of Uno!

Sample Proof

• Begin with

• List how to play these cards

• To get to

Given

Same Color Change Color Same Color Same Color

Same Color

You don’t have to use every postulate you know in every

proof.

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. Given2. Blue Skip ▐ 2. Same Color3. Wild Draw 4 ▐ 3. Change Color4. Yellow 5 ▐ 4. Same Color5. Yellow 1 ▐ 5. Same Color6. Yellow Reverse ▐ 6. Same Color

Given: Prove:

Using:


Given: Blue 5 Prove: Green 6

Statements (What Comes Next): Reasons (I can play this card because):

----------------------------------------------------------1. Blue 5 ▐ 1. Given2. Blue 1 ▐ 2. Same Color3. Green 1 ▐ 3. Same Number4. Green 6 ▐ 4. Same Color

Given: Prove:

Using:


Given: Blue Seven Prove: Red Nine

Statements (What Comes Next): Reasons (I can play this card because ):

---------------------------------------------------------------------1. Blue 7 ▐ 1. Given2. Green 7 ▐ 2. Same Number3. Green 4 ▐ 3. Same Color4. Yellow 4 ▐ 4. Same Number5. Yellow 9 ▐ 5. Same Color6. Red 9 ▐ 6. Same Number

Given: Prove:

Using:


Given: Red Reverse Prove: Green Nine

Statements (What Card to Play): Reasons (Why I can play the card):

---------------------------------------------------------------------1. Red Reverse ▐ 1. Given2. Red 3 ▐ 2. Same Color3. Blue 3 ▐ 3. Same Number4. Wild ▐ 4. Change Color5. Green 5 ▐ 5. Same Color6. Green 9 ▐ 6. Same Color

Given: Prove:

Using:


Given: Yellow Skip Prove: Blue 3

Statements (What Card to Play): Reasons (I can play this card because):

---------------------------------------------------------------------1. Yellow Skip ▐ 1. Given2. Yellow 8 ▐ 2. Same Color3. Red 8 ▐ 3. Same Number4. Green 8 ▐ 4. Same Number5. Blue 8 ▐ 5. Same Number6. Blue 3 ▐ 6. Same Color

Given: Prove:

Using:


Given: Yellow 8 Prove: Blue 1

Statements: Reasons:---------------------------------------------------------------------1. Yellow 8 ▐ 1. Given2. Yellow Skip ▐ 2. Same Color3. Green Skip ▐ 3. Same Number4. Green Draw 2 ▐ 4. Same Color5. Red Draw 2 ▐ 5. Same Number6. Red 5 ▐ 6. Same Color7. Blue 5 ▐ 7. Same Number8. Blue 1 ▐ 8. Same Color

How to do a Proof

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