WARM UP EXERCSE Consider the right triangle below with M the midpoint of the hypotenuse. Is MA = MC? Why or why not? MC B A 1.

WARM UP EXERCSEConsider the right triangle below with M the midpoint of the hypotenuse. Is MA = MC? Why or why not?

M

C

BA

1

WARM UP EXERCSEConsider the right triangle below with M the midpoint of the hypotenuse. Is MA = MC? Why or why not?

M

C

BA

2

3

§1.1 Introductory Material

The student will learn about:

math systems,

basic axioms, and geometric proof.

basic terms,

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§ 1.1 Statements and Reasoning

Reasoning – Learning Geometry Requires Time, Vocabulary Development, Attention to Detail and Order, Supporting Claims, and a Lot of Thinking. The Following Types of Thinking or Reasoning Are Used to Develop Mathematical Principles

Types of Reasoning

Intuition – an inspiration leading to the statement of a theory.


Induction – an organized effort to test the theory.


Induction – an organized effort to test the theory.

Deduction – A formal argument that proves the tested theory.

Mathematical System. A mathematical system consist of:

• Undefined terms.



• Defined terms.



• Defined terms.

• Axioms and postulates.



• Defined terms.

• Axioms and postulates.

• Theorems.

“With postulates, my dear, you need a gentle touch, They should not say too little, they should not say too much,And on one point above all, we must be insistent,Though postulates need not be ‘true,’ there set must be consistent.” Journey into Geometries by Marta Sved

Example

Axiom 1: Through any two distinct points there is exactly one line.

Design a geometry that fits these postulates.

Axiom 2: Every line has at least two distinct points.

Axiom 3: Not all points are on one line.

Example

Points – Apricot, Banana and Chocolate.

Lines – Apricot-Banana, Apricot-Chocolate and Banana-Chocolate.

Axiom 1: Through any two distinct points there is exactly one line.

Axiom 2: Every line has at least two distinct points.

Axiom 3: Not all points are on one line.

Axiomatic Systems - Example

Axiom 1: Every line contains at least two points. Axiom 2: Each two lines intersect in a unique point.Axiom 3: There are precisely three lines.

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A

B

C

A model.

“What is thinking? I should have thought I would have known.”

– Karl Gerstner

Conditional Statements

A conditional statement is written in the form, If p then q, or p implies q, and is symbolized by p → q. The condition p is called the hypothesis and q is the conclusion.

If p is a statement then ~ p is the negation of statement p.

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A conditional statement is in the form, p → q.

Converse: q → p.

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You should be familiar with the converse, inverse and contrapositive of a statement.

Inverse: then ~ p → ~ q.

Contrapositive: ~ q → ~ p.

“You should say what you mean.” the March hare went on.“I do,” Alice hastily replied; “At least – at least I mean what I say – that’s the same thing , you know.”“not the same thing a bit!” said the Hatter. “Why, you might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”

Alice in Wonderland by Lewis Carroll


Find the conditional statements and any converses, inverses, or contrapositives.

Converse: q → p.

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Inverse: then ~ p → ~ q. Contrapositive: ~ q → ~ p.



Find the conditional statements and any converses, inverses, or contrapositives.

Converse: q → p.

16

Inverse: then ~ p → ~ q.

Contrapositive: ~ q → ~ p.


Valid Arguments

An argument is valid if when all the premises are true then the conclusion is true.In a logic class truth tables are used to prove arguments valid.

Can you do that with the previous statements from Alice and Wonderland?

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In this class we will use the historically proven methods of proof to arrive at valid conclusions.

Types of Reasoning

Direct Proof

A Formal Proof Consist of the Following:

1. A statement or statements of what is given.

2. A statement of what is to be proven.

3. A drawing.

4. The proof in two column or paragraph form.

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Example of Direct ProofVertical Angle Theorem

Given: Intersecting lines l and m. l1

m 3

42

Statement Reason

l is a straight line Given

m is a straight line Given

m 1 + m 2 = 180 Def straight line.

m 2 + m 4 = 180 Def straight line.

m 1 + m 2 = m 2 + m 4 Arithmetic axiom

m 1 = m 4 Subtraction

QED or W 5 19

Prove: m 1 = m 4.

Types of Reasoning

Indirect Proof

An indirect proof should have the same four parts of a direct proof. The indirect proof assumes the conclusion is false and arrives at a contradiction to what is given. This method is sometimes referred to as “reductio ad absurdum”.

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Types of Reasoning

Indirect Proof

Indirect proof works particularly well when:

The negation of the initial premise P is easy.

When Q contains a negation and denies some claim.

Existence theorems.

Uniqueness theorems.

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Example of Indirect Proof

Vertical Angle Theorem

Given: Intersecting lines l and m.

Prove: m 1 = m 4. l

1

m 3

42

Statement Reason

1. m1 ≠ m4 Assumed

2. m 1 + m 2 ≠ m 4 + m 2 Arithmetic

3. m 1 + m 2 = 180 Def straight line.


→ ← 3 & 4 Contradict #2

m 1 = m 4 Assumption false

QED or W 5 22

Types of Reasoning

Proof by Elimination/Exhaustion

An elimination proof should have the same four parts of a direct proof. It is useful when there are finite possible events that occur and you can eliminate all but one of them. Then the remaining event must occur.

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Example of Elimination Proof



Prove: m 1 = m 4. l

1

m 3

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Statement Reason

1. Either m1 < m4 or

m1 > m4 or m1 = m4

Mathematical trichotomy

2. m 1 < m 4 Assumption.

3. m 1 + m 2 < m 4 + m 2 Arithmetic



24Continued

6. → ← Contradiction 4 & 5 with 3

Example of Elimination Proof



Prove: 1 = 4. l

1

m 3

42

Statement Reason

7. Let m 1 > m 4 Assumption.

8. Use the previous argument 2 – 6 to arrive at a contradiction. 1 = 4 Remaining case

QED or W 5

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Types of Reasoning

There is one more type of proof we will use and that is called induction

1. Prove for the case where n = 1.

2. Assume it is true for the case where n = k.

3. Prove for the case where n = k + 1.

This idea is like a row of dominos falling after you knock over the first one.

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Example of Induction


Continued

n(n 1)Pr ove : 1 2 3 . . . n

2

1(1 1)1 1

2


k (k 1)Assume : 1 2 3 .. . k

2

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(k 1)(k 2)Pr ove : 1 2 3 .. . k (k 1)

2

2k 3k 2(k 1) From step 2.

2

k(k 1)

2

2 2k k 2k 2 k 3k 2Common denominator.

2 2 2

2 2k 3k 2 k 3k 2Addition of fractions.

2 2

1 2 3 .. .(k 1)(k 2)

Pr ove : (k 1)2

k

28

k (k 1)1 2 3 .. . k

2

Greek Proof

n(n 1)Pr ove : 1 2 3 . . . n

2

29

n

n + 1



Continued

2Please pr ove : 1 3 5 . . . 2n 1 n

21 1


2Assume : 1 3 5 .. . 2n 1 n

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22 (2k 1) (k 1) From ste 2.k p

2 2k 2k 1 k 2k 1 Binomial expansion.

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21 2 3 .. . 2k 1 k

2Pr ove : 1 3 . . . 2k 1 2k 1 k 1 2

1 3 .. . 2k 1Pr ove : 2k 1 k 1

Greek Proof

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n

n

2Pr ove : 1 3 5 . . . 2n 1 n

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Summary.

• We learned about direct proof.

• We learned about conditional statements.

• We learned about several types of reasoning.

• We learned about valid arguments.

• We learned about indirect proof.

• We learned about proof by exhaustion.

• We learned about proof by induction.

Assignment: §1.1

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Example

1. Through any two distinct points there is exactly one line.

2. Every line has at least two distinct points.

3. Not all points are on one line.

Points – Apricot, Banana and Chocolate.

Lines – Apricot-Banana, Apricot-Chocolate and Banana-Chocolate.

WARM UP EXERCSE Consider the right triangle below with M the midpoint of the hypotenuse. Is MA = MC? Why or why not? MC B A 1.

Documents

mathematical system

undefined terms

distinct points

marta sved slide

karl gerstner slide

basic terms

mathematical principles

example points apricot