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1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 Section 1.4 Proving Conjectures: Deductive Reasoning May 910:15 AM 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. To prove a conjecture is true for all cases, we use deductive reasoning. Drawing a specific conclusion Deductive Reasoning: through logical reasoning by starting with general assumptions that are know to be valid. Sep 1212:56 PM 1 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 3 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). *Will be done in chapter 2* Sep 121:13 PM 1. Venn Diagrams An illustration that uses overlapping or non-overlapping circles to show the relationship between groups of things. 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. Sep 121:17 PM 2 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 Examples: 1. All dogs are mammals. All mammals are vertebrates. Shaggy is a dog. What can be deduced about Shaggy? 2. Casey voted in the last election. Only people over 18 years old vote. What can be concluded about Casey? 3. 4. 5. Mammals have fur (or hair). Lions are classified as mammals. What can be deduced about lions? Jun 1212:25 PM Sep 121:25 PM 3 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 Sep 121:21 PM 2. Number Theory This strategy involves choosing a variable or variables to algebraically represent a situation. Even Integers: 2n, where n is an integer. Odd Integers: 2n + 1, where n is an integer. Consecutive Integers: n, n + 1, n + 2, etc, where n is an integer. Consecutive Odd Integers: 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: 2n, 2n + 2, 2n + 4, etc, where n is an integer. Sep 123:42 PM 4 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 Write an expression for the following: 1) an even number 2) an odd number 3) the sum of two consecutive even integers 4) the sum of the squares of two consecutive integers 5) The difference of squares of two consecutive integers. 6) a two digit number 7) The product of two consecutive integers . 8) The product of an even and an odd integer. Sep 1712:48 PM Using deductive reasoning to prove your conjecture Three Steps: 1. Define the variables (let n equal...). 2. Use algebra to prove the conjecture. 3. State what you have proven. Sep 128:02 PM 5 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 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. 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 Try a sample with larger numbers: 1233 + 1234 + 1235 + 1236 + 1237 = 6185 1235 x 5 = 6175 Jun 1212:16 PM How can you prove that Jon's conjecture is true for all integers? 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. Sep 121:05 PM 6 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 Reflect: Jon used inductive reasoning to develop his conjecture. We used deductive reasoning to prove Jon‛s conjecture. Jun 1212:19 PM Example: Conjecture: If you multiply two even integers then the product will be even. 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 128:01 PM 7 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 Example: Conjecture: If you multiply two odd integers then the product will be odd. 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 128:08 PM For each example support the conjecture inductively(showing three examples) and then prove it deductively: a) Conjecture: The sum of four consecutive integers is equivalent to the first and last integers added, then multiplied by two. Sep 128:21 PM 8 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 b) Conjecture: The sum of four consecutive integers is equivalent to the first and last integers added, then multiplied by two. Sep 2110:25 AM c) Conjecture: The square of the sum of two positive integers is greater than the sum of the squares of the same two integers. Sep 128:22 PM 9 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 Example: 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? Sep 128:40 PM 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. Deductive Let x = a number Sep 128:40 PM 10 1.4_aProvingConjectures(DeductiveReasoning).notebook October 10, 2013 Inductive Deductive Let x = a number Sep 128:43 PM Text pg. 31-33 #'s 1-11, 13, 14, 16,17, 19, 20 Mid-Chapter Review: page 35 #'s 2, 5, 8-11 Jun 1212:29 PM 11