1 / 80

Logic

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. Logic .

jalia
Download Presentation

Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Logic

  2. 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

  3. Short sweet and to the point

  4. Logic A crocodile steals a son from his father, and promises to return the child if the father can correctly guess what the crocodile will do. What happens if the father guesses that the child will not be returned to him? • ANSWER: There is no solution. • If the crocodile keeps the child, he violates his rule, as the father predicted correctly. • If the crocodile returns the child, he still violates his rule as the father’s prediction was wrong.

  5. Number Pattern Is this proof of how numbers were developed?

  6. Mathematical Proof 2 = 1 a = b a2 = ab a2 - b2 = ab-b2 (a-b)(a+b) = b(a-b) a+b = b b+b = b 2b = b 2 = 1

  7. 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.

  8. Inductive Reasoning • A form of reasoning that draws a conclusion based on the observation of patterns. • Steps Identify a pattern Make a conjecture • Find counterexample to disprove conjecture

  9. 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

  10. Inductive Reasoning Not Proof

  11. 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.

  12. Identifying a Pattern Find the next item in the pattern. 4, 9, 16, … Sums of odd numbers make up the pattern. 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 The next number is 25.

  13. The next figure is . Identifying a Pattern Find the next item in the pattern. In this pattern, the figure rotates 90° counter-clockwise each time.

  14. 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.

  15. 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.

  16. 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.

  17. 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

  18. Finding a Counterexample Show that the conjecture is false by finding a counterexample. For every integer n, n3 is positive. Pick integers and substitute them into the expression to see if the conjecture holds. Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds. Let n = –3. Since n3 = –27 and –27  0, the conjecture is false. n = –3 is a counterexample.

  19. Inductive Reasoning • Example 1 • 90% of humans are right-handed. • Joe is a human. • Example 2 • Every life form that everyone knows of depends on liquid water to exist. • Example 3 • All of the swans that all living beings have ever seen are white. Therefore, the probability that Joe is right-handed is 90%. • Therefore, all known life depends on liquid water to exist. Therefore, all swans are white. • Inductive reasoning allows for the possibility that the conclusion is false, • even where all of the premises are true

  20. Conjectures about our class….

  21. Logic Common Sense

  22. 23° 157° True? Supplementary angles are adjacent. The supplementary angles are not adjacent, so the conjecture is false.

  23. Look at the triangle of six pennies below:I want to turn this triangle upside-down: So that it looks like this: What is the smallest number of coins I must move? Two coins

  24. A ship has a rope ladder hanging over its side. The rungs are 230 mm apart (23 cm). How many rungs will be underwater when the tide has risen one metre? None of the rungs go underwater. WHY?

  25. What mathematical symbol can be put between 5 and 9, to get a number bigger than 5 and smaller than 9? A Decimal Point 5.9

  26. Homework 2.1 and 2.2

  27. If…

  28. 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.

  29. Underline the hypothesis twice • The conclusion once • A figure is a parallelogram if it is a rectangle. • Four angles are formed if two lines intersect.

  30. 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.

  31. 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.

  32. For Problems 1 and 2: Identify the hypothesis and conclusion of each conditional. 1. A triangle with one right angle is a right triangle. 2. All even numbers are divisible by 2. 3. Determine if the statement “If n2 = 144, then n = 12” is true. If false, give a counterexample. H: A triangle has one right angle. C: The triangle is a right triangle. H: A number is even. C: The number is divisible by 2. False; n = –12.

  33. Identify the hypothesis and conclusion of each conditional. 1.A mapping that is a reflection is a type of transformation. 2.The quotient of two negative numbers is positive. 3. Determine if the conditional “If x is a number then |x| > 0” is true. If false, give a counterexample. H: A mapping is a reflection. C: The mapping is a transformation. H: Two numbers are negative. C: The quotient is positive. False; x = 0.

  34. 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, thenit does not have 4 paws. There are animals that are not cats that have 4 paws, so the inverse is false. Contrapositive: If an animal does not have 4 paws, thenit is not a cat; True. Cats have 4 paws, so the contrapositive is true.

  35. 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.

  36. 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.

  37. Conditional Statements Transition from hypothesis to p from conclusion to q Facilitates analysis of both parts.

  38. Statement Pythagoras is Greek. A triangle has three sides. Paris is the capital of Spain.

  39. Writing Conditional Statements Given p: You give me twenty dollars. q: I will be your best friend. Write the following statement in logic notation "If you give me $20, then I will be your best friend" p ->q q -> p What of: If you are my friend, I will give you $20.

  40. Truth Values of Conditional Statements If today is Friday, then tomorrow is Saturday. If the month is October, then next month is November. If you are 14 years old, then at your next birthday you will be 15 years old. If you are 13 years old, then at your next birthday you are eligible to receive a permit to drive. If you If you are in Mrs. Kapler’s Math class, you have homework for tonight. If you live in a pineapple under the sea, then you are Sponge Bob. If you are Sponge Bob, then you live in a pineapple under the sea. If you are cold, then you will put on a sweatshirt. If you did not eat breakfast, then you are hungry. If you ate breakfast, then you are not hungry. If you attend Washington Technology, then you are in 8th grade. If you are in 8th grade, then you attend Washington Technology. p – proposition q – next letter • p q p->q • T TT • T F F • F T T • F F T

  41. Truth Values of Conditional Statements If today is Friday, then tomorrow is Saturday. If the month is October, then next month is November. If you are 14 years old, then at your next birthday you will be 15 years old. If you are 13 years old, then at your next birthday you are eligible to receive a permit to drive. If you play basketball, then you are as tall as Mr. Lott. If you are in Mrs. Kapler’s Math class, you have homework for tonight. If you live in a pineapple under the sea, then you are Sponge Bob. If you are Sponge Bob, then you live in a pineapple under the sea. If you are cold, then you will put on a sweatshirt. If you did not eat breakfast, then you are hungry. If you ate breakfast, then you are not hungry. If you attend Washington Technology, then you are in 8th grade. If you are in 8th grade, then you attend Washington Technology. If you are in Geometry, then you are in 8th grade.

  42. Truth Values of Conditional Statements • p q p->q • T TT • T F F • F T T • F F T Given p: x is prime q: x is odd What is the truth value of a -> q when x = 2? 2. Given p: x is prime q: x is odd What is the truth value of p -> q when x = 9? False True

  43. Conditional (if, then) statements • If you give a mouse a cookie, then he will need a glass of milk. • If you give him a glass of milk, then he will ask for a straw. • I have no straws. Should I give the mouse a cookie?

  44. Conditional Statements handout • If we have class outside today, we will need to wear jackets and sweatshirts. • If you have English next hour, then you will go to the English classroom. • If you don’t eat breakfast in the morning, you will be hungry in your morning classes. • If your teacher is an alien, then he has green skin. • If Mrs. Kapler retired last year, then she continued teaching. • If today is Friday, then tomorrow is Wednesday.

  45. Converse, Inverse, Contrapositive • Conditional statement: p -> q • Converse: q -> p • Inverse: ~p -> ~q • Contrapositive: ~q -> ~p • Which statements are logically equivalent?

  46. Homework 2.3

  47. To determine truth in geometry… Beyond a shadow of a doubt. Deductive Reasoning.

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

More Related