Logic

Logic

• • • • Logical progression of thought A path others can follow and agree with Begins with a foundation of accepted In Euclidean Geometry begin with point, line and plane

Short sweet and to the point

Number Pattern Is this proof of how numbers were developed?

Mathematical Proof

a

2

- b 2 = 1 a

2

a = b = ab

2

= ab-b

2

(a-b)(a+b) = b(a-b) a+b = b b+b = b 2b = b 2 = 1

Geometry

• • Undefined terms Are not defined, but instead explained.

Form the foundation for all definitions in geometry.

• Postulates A statement that is accepted as true without proof. • Theorem A statement in geometry that has been proved.

Inductive Reasoning

• A form of reasoning that draws a conclusion based on the observation of patterns.

• Steps 1. Identify a pattern 2. Make a conjecture • Find counterexample to disprove conjecture

Inductive Reasoning

• Does not definitely prove a statement, rather assumes it • Educated Guess at what might be true Example Polling 30% of those polled agree therefore 30% of general population

Inductive Reasoning Not Proof

Identifying a Pattern

Find the next item in the pattern.

7, 14, 21, 28, …

Multiples of 7 make up the pattern.

The next multiple is 35.

Identifying a Pattern

Find the next item in the pattern.

4, 9, 16, …

Sums of odd numbers make up the pattern.

The next number is 25

.

1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25

1 2 = 1 2 2 = 4 3 2 = 9 4 2 = 16 5 2 = 25

Identifying a Pattern Find the next item in the pattern.

In this pattern, the figure rotates 90° counter-clockwise each time.

The next figure is .

Making a Conjecture Complete the conjecture.

The sum of two odd numbers is ? .

List some examples and look for a pattern.

1 + 1 = 2 3.14 + 0.01 = 3.15

3,900 + 1,000,017 = 1,003,917

The sum of two positive numbers is positive.

Identifying a Pattern Find the next item in the pattern.

January, March, May, ...

Alternating months of the year make up the pattern.

The next month is July. The next month is… then August

Perhaps the pattern was… Months with 31 days.

Complete the conjecture.

The product of two odd numbers is ? .

List some examples and look for a pattern.

1



1 = 1 3



3 = 9 5



7 = 35

The product of two odd numbers is odd.

Inductive Reasoning

Counterexample - An example which disproves a conclusion • Observation 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 are odd • Conclusion All prime numbers are odd.

2

is a counterexample

Finding a Counterexample Show that the conjecture is false by finding a counterexample.

For every integer n, n 3 is positive.

Pick integers and substitute them into the expression to see if the conjecture holds. Let n = 1. Since n 3 = 1 and 1 > 0, the conjecture holds. Let n = –3. Since n 3 = –27 and –27  0, the conjecture is false.

n = –3 is a counterexample.

Inductive Reasoning

Example 1 90% of humans are right-handed.

Joe is a human.

Therefore, the probability that Joe is right-handed is 90%.

Example 2 Every life form that everyone knows of depends on liquid water to exist.

Therefore, all known life depends on liquid water to exist.

Example 3 All of the swans that all living beings have ever seen are white.

Therefore, all swans are white.

Inductive reasoning allows for the possibility that the conclusion is false, even where all of the premises are true

Conjectures about our class….

True?

Supplementary angles are adjacent.

23 ° 157 ° The supplementary angles are not adjacent, so the conjecture is false.

Homework 2.1 and 2.2

To determine truth in geometry… Information is put into a conditional statement.

The truth can then be tested.

A conditional statement in math is a statement in the if-then form.

If hypothesis, then conclusion A bi-conditional statement is of the form If and only if. If and only if hypothesis, then conclusion.

Underline the hypothesis twice The conclusion once 1. A figure is a parallelogram if it is a rectangle.

2. Four angles are formed if two lines intersect.

Analyzing the Truth Value of a Conditional Statement Determine if the conditional is true. If false, give a counterexample.

If two angles are acute, then they are congruent.

You can have acute angles with measures of 80 ° and 30 ° . In this case, the hypothesis is true, but the conclusion is false.

Since you can find a counterexample, the conditional is false.

Analyzing the Truth Value of a Conditional Statement Determine if the conditional is true.

“If a number is odd, then it is divisible by 3” If false, give a counterexample.

An example of an odd number is 7. It is not divisible by 3. In this case, the hypothesis is true, but the conclusion is false. Since you can find a counterexample, the conditional is false.

For Problems 1 and 2: Identify the hypothesis and conclusion of each conditional.

1. A triangle with one right angle is a right triangle. H: A triangle has one right angle. C: The triangle is a right triangle.

2. All even numbers are divisible by 2.

3.

H: A number is even. C: The number is divisible by 2.

Determine if the statement “If n 2 = 144, then n = 12” is true. If false, give a counterexample.

False;

n

= – 12.

Identify the hypothesis and conclusion of each conditional.

1.

A mapping that is a reflection is a type of transformation.

H: A mapping is a reflection.

C: The mapping is a transformation.

2.

The quotient of two negative numbers is positive.

H: Two numbers are negative.

C: The quotient is positive.

3.

Determine if the conditional “If x is a number then |x| > 0” is true.

If false, give a counterexample.

False; x = 0.

Different Forms of Conditional Statements

Given Conditional Statement If an animal is a cat , then it has four paws .

Converse: If an animal has 4 paws , then it is a cat. There are other animals that have 4 paws that are not cats, so the converse is false.

Inverse: If an animal is not a cat, then have 4 paws.

it does not There are animals that are not cats that have 4 paws, so the inverse is false.

Contrapositive: then If an animal does not have 4 paws, it is not a cat; True.

Cats have 4 paws, so the contrapositive is true.

A bi-conditional statement is of the form If and only if. If and only if hypothesis, then conclusion.

Example A triangle is isosceles if and only if the triangle has two congruent sides.

Write as a biconditional Parallel lines are two coplanar lines that never intersect

Two lines are parallel if and only if they are coplanar and never intersect.

Homework 2.3

To determine truth in geometry…

Deductive Reasoning.

Beyond a shadow of a doubt.

Deductive reasoning

Uses logic to draw conclusions from • Given facts • Definitions • Properties.

True or False And how do you know?

A pair of angles is a linear pair.

The angles are supplementary angles.

Two angles are complementary and congruent.

The measure of each angle is 45 .

Modus Ponens

Most common deductive logical argument

p

⇒

q p

∴

q If

p

, then

q p

, therefore

q

Example

If I stub my toe, then I will be in pain.

I stub my toe.

Therefore, I am in pain.

Modus Tollens

Second form of deductive logic is

p ~q

⇒ ∴

q ~p If

p

, then

q not q

, therefore

not p

Example

If today is Thursday, then the cafeteria will be serving burritos.

The cafeteria is not serving burritos, therefore today is not Thursday.

If-Then Transitive Property

Third form of deductive logic A chains of logic where one thing implies another thing.

p q

⇒ ⇒

r q

∴

p

⇒

r If

p

, then

q If

q

, then

r

, therefore if

p

, then

r

Example

If today is Thursday, then the cafeteria will be serving burritos.

If the cafeteria will be serving burritos, then I will be happy.

Therefore, if today is Thursday, then I will be happy.

Deductive reasoning

Three forms

p

⇒

q p

∴

q p

⇒

q p

⇒

q ~q q

⇒ ∴

~p r

∴

p

⇒

r

Draw a conclusion from the given information.

If a polygon is a triangle, then it has three sides.

If a polygon has three sides, then it is not a quadrilateral. Polygon P is a triangle.

Conclusion: Polygon P is not a quadrilateral.

Homework 2.4

Proof 1. Algebraic 2. Geometric

Proof - argument that uses • Logic • Definitions • Properties, and • Previously proven statements to show that a conclusion is true.

An important part of writing a proof is giving justifications to show that every step is valid.

Algebraic Proof

• Properties of Real Numbers Equality Distributive Property

a

(

b

+

c

) =

a b

+

a c

.

•

Substitution

Practice Solving an Equation with Algebra

Solve the equation 4 m – 8 = – 12. Write a justification for each step.

4 m – 8 = – 12 Given equation +8 +8 Addition Property of Equality 4

m

= – 4 Simplify.

Division Property of Equality

m

= – 1 Simplify.

Practice Solving an Equation with Algebra

Solve the equation . Write a justification for each step.

Given equation

t

= – 14 Multiplication Property of Equality.

Simplify.

Solving an Equation with Algebra

Solve for x. Write a justification for each step.

NO

=

NM

+

MO

4 x – 4 = 2

x

+ (3 x – 9)

Segment Addition Post.

Substitution Property of Equality

4 x – 4 = 5 x – 9 – 4 = x – 5 =

x

9

Simplify. Subtraction Property of Equality Addition Property of Equality

Homework 2.5

Algebraic Proof

Geometric Proof

  Prove geometric theorems by using deductive reasoning.

Two-column proofs.

Remember!

Numbers are equal (=) and figures are congruent (



).

When writing a proof: 1. Justify each logical step with a reason. 2. Each step must be clear enough so that anyone who reads your proof will understand them.

Hypothesis • • • • Definitions Postulates Properties Theorems Conclusion

Proof Steps: 1. Start with given (hypothesis) 2. Logically connect given to conclusion Progressives statements with reasons 3. End with conclusion Two Column Proof – organizes your work Statement Reason

Writing Reasons Using a Two Column Proof Write a reason for each step, given that



A

and



B

are supplementary and m



A

= 45 ° .

1.



A

and 

B

m 

A

= 45 ° are supplementary.

2. m 

A

+ m 

B

= 180 ° 3. 45 ° + m 

B

= 180 ° 4. m 

B

= 135 ° Given Def. of supp  s Subst. Prop of =

Steps 1, 2

Subtr. Prop of =

Writing Reasons Using a Two Column Proof Write a reason for each step, given that

B

is the midpoint of

AC

and

AB



EF

.

1.

B

is the midpoint of

AC

.

2.

AB



BC

3.

AB



EF

4.

BC



EF

Given Def. of mdpt.

Given Trans. Prop. of 

Completing a Two-Column Proof Given:

XY

Prove:

XY



XY

Statements 1.

XY

2.

XY = XY

3. .



XY

Reasons

1. Given

2. .

3. Def. of  segs.

Completing a Two-Column Proof Given:

 1 and  2 are supplementary, and  1   3

Prove:

 3 and  2 are supplementary. Example 4

Example 4 Continued Statements 1.

1 and  2 are supplementary.

 1   3

2.

m  1 + m  2 = 180 °

Reasons 1.

Given

2. .

 s

3. .

 1 = m  3

3.

Def. of   s

4.

m  3 + m  2 = 180 °

4.

Subst.

5.

 3 and  2 are supplementary

5.

Def. of supp.  s

Use a Two Column Proof Given:

 1,  2 ,  3,  4 Prove: m  1 + m  2 = m  1 + m  4

1.

 1 and  2 are supp.

 1 and  4 are supp.

2. m  1 + m  2 = 180 ° , m  1 + m  4 = 180 ° 1. Linear Pair Thm.

2. Def. of supp. 3. m  1 + m  2 = m  1 + m  4 3. Subst.  s

Homework 2.6

Geometric Proof

There are nine compositions (A to I) of eight colored cubes. Find two identical compositions. They can be rotated.

Solution:

Compositions D and I are identical.

Four flat cubes Their patterns are drawn with bold black lines. Which can be drawn without taking your pencil off the paper or going along the same line twice? Which of them can't be drawn in this way?

Shapes A and D can be drawn without taking your pencil off the paper or going along the same line twice. Shapes B and C can't be drawn in this way.