Section 1.4 Proving Conjectures: Deductive Reasoning May 9 10:15 AM 1
Definition: Proof: A mathematical argument showing that a statement is valid in all cases, or that no counterexample exists. Generalization: A principle, statement or idea that has general application. Sep 12 12:56 PM 2
Example 1: (Text page 27) Jon discovered a pattern when adding consecutive integers: a) 1 + 2 + 3 + 4 + 5 =15 b) (-15) + (-14) + (-13) + (-12) + (-11) = -65 c) (-3) + (-2) + (-1) + 0 + 1 = -5 He claims that whenever you add five consecutive integers, the sum is always 5 times the median of the numbers. Prove that Jon s Conjecture is true for all integers. Jun 12 12:16 PM 3
Definition: Median: the "middle number" (in a sorted list of numbers). To find the Median, place the numbers you are given in value order and find the middle number. Consecutive Numbers: Numbers which follow each other in order, without gaps, from smallest to largest. 12, 13, 14 and 15 are consecutive numbers. Sep 12 1:00 PM 4
Check Jon s examples: a) 1 + 2 + 3 + 4 + 5 =15 Median # = 3 x 5 = 15 b) (-15) + (-14) + (-13) + (-12) + (-11) = -65 Median # = -13 x 5 = -65 c) (-3) + (-2) + (-1) + 0 + 1 = -5 Median # = -1 x 5 = -5 Jun 12 12:17 PM 5
Try a sample with larger numbers: 1233 + 1234 + 1235 + 1236 + 1237 = 6185 1235 x 5 = 6175 Jun 12 12:17 PM 6
How can you prove that Jon's conjecture is true for all integers? To prove a conjecture is true for all cases, we use deductive reasoning. Definition: Deductive Reasoning: Drawing a specific conclusion through logical reasoning by starting with general assumptions that are know to be valid. Sep 12 1:05 PM 7
Prove using generalizations: Let x = the first number x + (x+1) + (x+2) + (x+3) + (x+4) = 5x + 10 = 5 (x+2) Since the sum equals five times the median number, Jon s conjecture is true. Jun 12 12:18 PM 8
If you can prove Jon s conjecture using generalizations, then you can prove it is true for all integers. Reflect: We used deductive reasoning to prove Jon s conjecture. Jon used inductive reasoning to develop his conjecture. Jun 12 12:19 PM 9
We will use three different strategies to prove conjectures. 1. Venn Diagrams (a visual representation) 2. Number Theory Proofs (choosing a variable to algebraically represent a situation). 3. Two-Column Proofs (using statements and reasons in an organized list). Sep 12 1:13 PM 10
Venn Diagrams - An illustration that uses overlapping or non-overlapping circles to show the relationship between groups of things. Sep 12 1:17 PM 11
Example: All zips are zaps. All zaps are zops. Shaggy is a zip. What can be deduced about Shaggy? Since Shaggy is a Zip, we also know he is a zap and a zop. Draw a Ven diagram. Jun 12 12:22 PM 12
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You can think of this example in different terms and it might help you to understand: All dogs are mammals. All mammals are vertebrates. Shaggy is a dog. What can be deduced about Shaggy? Shaggy is a dog, a mammal and a vertebrate. Jun 12 12:25 PM 14
Example: Hypothesis: Casey voted in the last election. Only people over 18 years old vote. Conclusion: Casey is over 18 years old. Sep 12 1:32 PM 15
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Example: Mammals have fur (or hair). Lions are classified as mammals. What can be deduced about lions? Illustrate your conjecture with a Venn diagram. Sep 12 3:44 PM 18
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Number Theory This strategy involves choosing a variable or variables to algebraically represent a situation. Example: Two consecutive integers: x and x + 1 Even integer: 2m Odd integer: 2n + 1 Sep 12 3:42 PM 22
Even Integers can be written in the form: 2m, where m is an integer. (Even numbers have a factor of two.) Odd Integers can be written in the form: 2n + 1, where n is an integer. (One more than an even number will always be an odd.) Consecutive Integers can be written in the form: n, n + 1, n + 2, etc, where n is an integer. (By adding one more to the previous number you will get the next consecutive integer.) Consecutive Odd Integers can be written in the form: 2n + 1, 2n + 3, 2n + 5, etc, where n is an integer. (By adding two more to the previous number you will get the next consecutive odd integer.) Consecutive Even Integers can be written in the form: 2n, 2n + 2, 2n + 4, etc, where n is an integer. (By adding two more to the previous number you will get the next consecutive even integer.) Sep 12 7:45 PM 23
Multiply: Sep 18 8:07 AM 24
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Write an expression for the following: 1) an even number Sep 17 12:48 PM 26
Write an expression for the following: 2) an odd number Sep 17 2:24 PM 27
Write an expression for the following: 3) the sum of two consecutive even integers Sep 17 2:24 PM 28
Write an expression for the following: 4) the sum of two positive consecutive odd integers Sep 17 2:25 PM 29
Write an expression for the following: 5) the sum of 5 consecutive natural numbers Sep 17 12:49 PM 30
Write an expression for the following: 6) the sum of three consecutive even integers Sep 17 2:26 PM 31
Write an expression for the following: 7) the sum of the squares of two consecutive integers Sep 17 2:26 PM 32
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Write an expression for the following: 8) The sum of the squares of two consecutive even integers (2 versions) Sep 17 2:26 PM 34
Write an expression for the following: 9) The sum of 7 consecutive integers Sep 17 2:17 PM 35
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Write an expression for the following: 10) The product of two consecutive integers. Sep 17 2:20 PM 37
Write an expression for the following: 11)The difference of squares of two consecutive integers. Sep 17 2:20 PM 38
Write an expression for the following: 12) a two digit Sep 17 2:20 PM 39
Write an expression for the following: 13) a three digit number Sep 17 2:20 PM 40
Write an expression for the following: 14) The sum of a two digit and a three digit number Sep 17 2:30 PM 41
Write an expression for the following: 15) The sum of a two digit number and the number formed by reversing its digits. Sep 17 2:30 PM 42
Write an expression for the following: 16) The sum of a three digit number and the number formed by reversing its digits. Sep 17 2:30 PM 43
Write an expression for the following: 17) The product of two consecutive integers. Sep 17 2:30 PM 44
Write an expression for the following: 18) The product of an even and an odd integer. Sep 17 2:30 PM 45
Write an expression for the following: 19) The square of an odd integer. Sep 17 2:30 PM 46
Inductive Reasoning and Deductive Proofs Sep 17 3:13 PM 47
What happens when you multiply two even integers? Conjecture: If you multiply two even integers then the product will be even. Sep 12 8:01 PM 48
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Now use deductive reasoning to prove your conjecture: Three Steps: 1. Define the variables. 2. Use algebra to prove the conjecture. 3. State what you have proven. Sep 12 8:02 PM 50
Conjecture: If you multiply two even integers then the product will be even. Sep 20 10:30 AM 51
Let 2m = one even integer Let 2n = a second even integer The product = 2m X 2n = 4mn = 2(2mn) By showing the product has a factor of 2 you are proving that it is even. Therefore, if you multiply two even integers then the product will be even. Sep 12 8:04 PM 52
What happens when you multiply two odd integers? Conjecture: If you multiply two odd integers then the product will be odd. Sep 12 8:08 PM 53
Conjecture: If you multiply two odd integers then the product will be odd. Sep 20 10:41 AM 54
Let 2m + 1 = one odd integer Let 2n + 1 = a second odd integer The product = (2m + 1) X (2n + 1) = 4mn + 2n + 2m + 1 = 2(2mn + n + m) + 1 By showing the product is 2 times an integer plus 1 you are proving that it is odd. Therefore, if you multiply two odd integers then the product will be odd. Sep 12 8:09 PM 55
Conjecture: The sum of four consecutive integers is equivalent to the first and last integers added, then multiplied by two. a) defend the conjecture inductively by showing 2 examples. b) prove the conjecture deductively. Sep 12 8:21 PM 56
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Conjecture: The sum of four consecutive integers is equivalent to the first and last integers added, then multiplied by two. Sep 21 10:25 AM 58
Support the following inductively(showing three examples) and then prove it deductively: The square of the sum of two positive integers is greater than the sum of the squares of the same two integers. Sep 12 8:22 PM 59
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Support the following inductively(showing three examples) and then prove it deductively: The sum of a 2-digit number and the number formed by reversing its digits will always be divisible by 11. Sep 12 3:46 PM 63
Support the following inductively(showing three examples) and then prove it deductively: The difference between an odd integer and an even integer is odd. Sep 12 3:47 PM 64
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Prove the following inductively(showing at least three examples) and then prove it deductively: An odd number times an even number is even. Inductive Deductive Let 2n represent an even number Let 2m + 1 be an odd number 2n(2m + 1) = 4nm + 2n = 2(2nm + n) Since 2(2nm + n) is divisible by 2, it is even. Sep 12 8:33 PM 67
Consider the following problem: Think of any number. Multiply that number by 2, then add 6, and divide the result by 2. Next subtract the original number. What is the result? (a) Use inductive reasoning to make a conjecture for the answer. (b) Use deductive reasoning to prove your conjecture. Sep 12 8:40 PM 68
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Inductive So we might form a conjecture that the result will always be the number 3. But this doesn t prove the conjecture, as we ve tried only two of infinitely many possibilities. Sep 12 8:40 PM 72
Deductive Let x = a number Sep 12 8:42 PM 73
Select a number. Add 50 to the number. Multiply the sum by 2. Subtract the original number from the product. What is the result? (a) Use inductive reasoning to arrive at a general conclusion. (b) Use deductive reasoning to prove your conclusion is true. Sep 12 8:42 PM 74
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Inductive Sep 12 8:43 PM 78
Deductive Let x = a number Sep 12 8:44 PM 79
Two-Column Proof Two column proofs will be studied in depth in chapter 2. The following will be an introductory example only. Sep 12 3:43 PM 80
Example: Prove that when two straight lines intersect, the vertically opposite angles are equal. Jun 12 12:26 PM 81
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1.4 Assignment: Nelson Foundations of Mathematics 11, Sec 1.4, pg. 31 33 Questions: 1 11, 13, 14, 16,17, 19, 20 Mid Chapter Review: page 35 #'s 2, 5, 8 11 Jun 12 12:29 PM 85