1.1 Inductive Reasoning filled in.notebook August 20, 2015
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1 1.1 Inductive Reasoning 1
2 Vocabulary Natural or Counting Numbers Ellipsis Scientific Method Hypothesis or Conjecture Counterexample 2
3 Vocabulary Natural or Counting Numbers 1, 2, 3, 4, 5... positive whole numbers Ellipsis Scientific Method Hypothesis or Conjecture Counterexample 3
4 Vocabulary Natural or Counting Numbers 1, 2, 3, 4, 5... positive whole numbers Ellipsis three dots indicating a continuation of the pattern Scientific Method Hypothesis or Conjecture Counterexample 4
5 Vocabulary Natural or Counting Numbers 1, 2, 3, 4, 5... positive whole numbers Ellipsis three dots indicating a continuation of the pattern Scientific Method the process for proving (or disproving) a hypothesis after observations of specific cases Hypothesis or Conjecture Counterexample 5
6 Vocabulary Natural or Counting Numbers 1, 2, 3, 4, 5... positive whole numbers Ellipsis three dots indicating a continuation of the pattern Scientific Method the process for proving (or disproving) a hypothesis after observations of specific cases Hypothesis or Conjecture a prediction based on specific observations Counterexample 6
7 Vocabulary Natural or Counting Numbers 1, 2, 3, 4, 5... positive whole numbers Ellipsis three dots indicating a continuation of the pattern Scientific Method the process for proving (or disproving) a hypothesis after observations of specific cases Hypothesis or Conjecture a prediction based on specific observations Counterexample a specific example that proves that the conjecture is false 7
8 Vocabulary Natural or Counting Numbers 1, 2, 3, 4, 5... positive whole numbers Ellipsis three dots indicating a continuation of the pattern Scientific Method the process for proving (or disproving) a hypothesis after observations of specific cases Hypothesis or Conjecture a prediction based on specific observations Counterexample a specific example that proves that the conjecture is false Rules about counterexamples: It takes only one to disprove a conjecture Not finding one neither proves or disproves a conjecture 8
9 Inductive vs. Deductive Reasoning 9
10 Inductive vs. Deductive Reasoning Inductive Reasoning is the process of reasoning to a general conclusion through observations of specific cases. 10
11 Inductive vs. Deductive Reasoning Inductive Reasoning is the process of reasoning to a general conclusion through observations of specific cases. Deductive Reasoning is the process of reasoning to a specific conclusion from a general statement 11
12 Inductive vs. Deductive Reasoning Inductive Reasoning is the process of reasoning to a general conclusion through observations of specific cases. Deductive Reasoning is the process of reasoning to a specific conclusion from a general statement The Process: Observe a trend Make a general conclusion based on the trend (IR) Make a hypothesis to prove Look for a counterexample If you can't find a counterexample make a proof (DR) If your proof holds up, then you have proven your hypothesis 12
13 When will you use inductive reasoning? When will you use deductive reasoning? 13
14 General Notation for Proofs: Odd numbers: (2n 1) Even numbers: (2n) **Use n for the first odd or even number, then m, then p, etc. **Always try to get your final number in the form of an odd or even number 14
15 Example 1: The product of two odd numbers Will the product of two odd numbers always be odd? 15
16 Example 2: The sum of an odd and an even number If an odd number and an even number are added, will the sum be an odd or an even number? 16
17 Example 3: Divisibility If the last two digits of a number are divisible by seven will the number thenbe divisible by seven? 17
18 Example 4: Prove or Disprove a) The difference of any two counting numbers will be a counting number. b) The product of any two counting numbers will be a counting number. 18
19 Example 5: Pick a number n. Pick any number, multiply the number by 4, add 6 to the product, divide the sum by 2, and subtract 3 from the quotient. What can we conclude? 19
20 Example 6: Pick a number n. Prove the conjecture from Example 5. Pick any number, multiply the number by 4, add 6 to the product, divide the sum by 2, and subtract 3 from the quotient. 20
21 IN YOUR HOMEWORK Pay close attention to #39 42!!! 21
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