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

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Logic

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

  5. 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

  6. 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.

  7. 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

  8. 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

  9. Inductive Reasoning Not Proof

  10. 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.

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. 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

  17. 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.

  18. 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

  19. Conjectures about our class….

  20. 23° 157° True? Supplementary angles are adjacent. The supplementary angles are not adjacent, so the conjecture is false.

  21. Homework 2.1 and 2.2

  22. 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.

  23. 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.

  24. 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.

  25. 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.

  26. 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.

  27. 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.

  28. 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.

  29. 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.

  30. 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.

  31. Homework 2.3

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

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

  34. 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 .

  35. 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.

  36. 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.

  37. If-Then Transitive Property • Third form of deductive logic • A chains of logic where one thing implies another thing. • p ⇒ q • q ⇒ r ∴ 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.

  38. Deductive reasoning • Three forms • p ⇒ q p ∴ q • p ⇒ q ~q ∴ ~p • p ⇒ q q ⇒ r ∴ p ⇒ r

  39. 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.

  40. Homework 2.4

  41. Proof Algebraic Geometric

  42. 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.

  43. Algebraic Proof • Properties of Real Numbers Equality Distributive Property a(b + c) = ab + ac. • Substitution

  44. Division Property of Equality Practice Solving an Equation with Algebra Solve the equation 4m – 8 = –12. Write a justification for each step. 4m – 8 = –12 Given equation +8+8Addition Property of Equality 4m = –4 Simplify. m = –1 Simplify.

  45. Solve the equation . Write a justification for each step. Given equation Multiplication Property of Equality. t = –14 Simplify. Practice Solving an Equation with Algebra

  46. Solving an Equation with Algebra Solve for x. Write a justification for each step. NO = NM + MO Segment Addition Post. 4x – 4 = 2x + (3x – 9) Substitution Property of Equality 4x – 4 = 5x – 9 Simplify. –4 = x – 9 Subtraction Property of Equality 5 = x Addition Property of Equality

  47. Homework 2.5 Algebraic Proof

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