Transcript 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 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. Van Hiele Levels Children at Level 0 will often say all of these shapes are triangles, except E, which is too "skinny". They may say F is "upside down". Students at Level 1 will recognize that only E and F are valid triangles Number Pattern Is this proof of how numbers were developed? 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 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. 1=1 The next number is 25. 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 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 positive 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. 11=1 33=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, 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. Example 1 Inductive Reasoning What can you conclude? 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…. Logic Common Sense True? Supplementary angles are adjacent. 23⁰ 157⁰ Based on the definition, supplementary angles sum to 180⁰. Therefore, supplementary angles can be non-adjacent, so the conjecture is false. Look at the triangle of six pennies below: I want to turn this triangle upside-down: Two coins So that it looks like this: What is the smallest number of coins I must move? 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? 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 Homework 2.1 and 2.2 If… 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. H: A number is even. C: The number is divisible by 2. 3. Determine if the statement “If n2 = 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 2. The a reflection is a type of transformation. H: A mapping is a reflection. C: The mapping is a transformation. 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 it 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, then 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. Conditional Statements Transition from hypothesis to p from conclusion to q Facilitates analysis of both parts. Statement Pythagoras is Greek. A triangle has three sides. Paris is the capital of Spain. 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. Truth Values of Conditional Statements p – proposition q – next letter • • • • • p T T F F q T F T F p->q T F T T 1. If today is Friday, then tomorrow is Saturday. 2. If the month is October, then next month is November. 3. If you are 14 years old, then at your next birthday you will be 15 years old. 4. If you are 13 years old, then at your next birthday you are eligible to receive a permit to drive. 5. If you 6. If you are in Mrs. Kapler’s Math class, you have homework for tonight. 7. If you live in a pineapple under the sea, then you are Sponge Bob. 8. If you are Sponge Bob, then you live in a pineapple under the sea. 9. If you are cold, then you will put on a sweatshirt. 10. If you did not eat breakfast, then you are hungry. 11. If you ate breakfast, then you are not hungry. 12. If you attend Washington Technology, then you are in 8th grade. 13. If you are in 8th grade, then you attend Washington Technology. Truth Values of Conditional Statements 1. If today is Friday, then tomorrow is Saturday. 2. If the month is October, then next month is November. 3. If you are 14 years old, then at your next birthday you will be 15 years old. 4. If you are 13 years old, then at your next birthday you are eligible to receive a permit to drive. 5. If you play basketball, then you are as tall as Mr. Lott. 6. If you are in Mrs. Kapler’s Math class, you have homework for tonight. 7. If you live in a pineapple under the sea, then you are Sponge Bob. 8. If you are Sponge Bob, then you live in a pineapple under the sea. 9. If you are cold, then you will put on a sweatshirt. 10.If you did not eat breakfast, then you are hungry. 11.If you ate breakfast, then you are not hungry. 12.If you attend Washington Technology, then you are in 8th grade. 13. If you are in 8th grade, then you attend Washington Technology. 14.If you are in Geometry, then you are in 8th grade. Truth Values of Conditional Statements • • • • • p T T F F q T F T F p->q T F T T 1. Given p: x is prime q: x is odd What is the truth value of a -> q when x = 2? False 2. Given p: x is prime q: x is odd What is the truth value of p -> q when x = 9? True 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? 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. Converse, Inverse, Contrapositive • • • • • Conditional statement: p -> q Converse: q -> p Inverse: ~p -> ~q Contrapositive: ~q -> ~p Which statements are logically equivalent? 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 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. Deductive reasoning Three forms p⇒q p∴q p⇒q ~q ∴ ~p p⇒q q⇒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) = ab + ac. • Substitution Practice Solving an Equation with Algebra Solve the equation 4m – 8 = –12. Write a justification for each step. 4m – 8 = –12 +8 +8 Given equation Addition Property of Equality 4m Simplify. = –4 Division Property of Equality m = –1 Simplify. Practice Solving an Equation with Algebra Solve the equation . Write a justification for each step. Given equation Multiplication Property of Equality. t = –14 Simplify. 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 –4 = x – 9 5=x 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 mA = 45°. 1. A and B are supplementary. mA = 45° Given 2. mA + mB = 180° Def. of supp s 3. 45° + mB = 180° Subst. Prop of = Steps 1, 2 Subtr. Prop of = 4. mB = 135° 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. Given 2. AB BC Def. of mdpt. 3. AB EF Given 4. BC EF Trans. Prop. of Completing a Two-Column Proof Given: XY Prove: XY XY Statements 1. XY 2. XY = XY 3. XY . XY Reasons 1. Given 2. Reflex. . Prop. of = 3. Def. of segs. Example 4 Completing a Two-Column Proof Given: 1 and 2 are supplementary, and 1 3 Prove: 3 and 2 are supplementary. Example 4 Continued Statements Reasons 1. 1 and 2 are supplementary. 1. Given 1 3 2. m1 + m2 = 180° of supp. s 2. Def. . = m3 3. m1 . 3. Def. of s 4. m3 + m2 = 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: m1 + m2 = m1 + m4 1. 1 and 2 are supp. 1. Linear Pair Thm. 1 and 4 are supp. 2. m1 + m2 = 180°, m1 + m4 = 180° 2. Def. of supp. s 3. m1 + m2 = m1 + m4 3. Subst. 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.