Inductive and Deductive Reasoning

• You should read this section.
• We want to talk about how we draw conclusions.
• Inductive Reasoning is the process of arriving at a general conclusion based on observations of specific examples
• A hypothesis or conjecture is a statement of a general principle based on specific observations.
• We do this all of the time.
  • It was cold yesterday, and the day before, therefore it will be cold today.
  • The sun was in the east this morning, overhead this afternoon and in the west this evening, therefore the sun is moving over the earth.
• These examples show the problem with inductive reasoning.
  • Insufficient observations can lead to incorrect conclusions.
  • Incorrect observations can lead to incorrect conclusions.
• Still, this is a major tool in science.
• The Scientific Method
  • Form a hypothesis
  • Test the hypothesis
  • Accept or reject the hypothesis
• A counterexample is a case where the hypothesis does not hold.
• Inductive reasoning often leads to investigation and discovery.
• For our purposes, we will use inductive reasoning to solve problems.
• An approach to homework:
  • The first example can be solved in a given method
  • A second example can be solved that way as well,
  • Perhaps all examples?
• Examples Of Homework type 1
  • (2) 19,, 24, 29, 34, 39, ___
    • (A) The number increases by 5 each time.
    • (B) 39 + 5 = 44
  • (18) 2, 5, 10, 17, 26, 37
    • (A) Starting with 3, add two more to the previous each time.
    • (A) 2+3 = 5, 5+5 = 10, 10+7 = 17, ...
    • (B) 37+13 = 50
  • (34)
    • (A) The first pattern is triangle square
    • (A) The second pattern is add 1 line each time.
    • (B) triangle with 5 lines
• Deductive reasoning is the process of proving a specific conclusion from one or more general statements
• This is how mathematicians prove things are true.
• When something is shown to be true this way, then it must be true, or the rules are incorrect.
• Show that the sum of two even integers is even.

  let a and b be even integers.
  Show a+b is even.
      a = 2i, b = 2j  where i and j are integers  (Def of even)
      a+b = 2i+2j                                 (Def of addition)
          = 2(i+j)                                (Distributive property)
  	which is even                           (Def of even)
     

• Example from homework
  • (36) Select a number, Multiply the number by 3. Add 6 to the product. Divide the Sum by 3. Subtract the original number form the quotient.
    • 0, (0*3+6)/3 - 0 = 2
    • 1, (1*3+6)/3 - 1 = 2
    • 2, (2*3+6)/3 - 2 = 2
    • 5, (5*3+6)/3 - 5 = 2
    • The result will be two.

    • (n*3+6)/3-n = 3(n+2)/3 - n
                  = n+2-n 
      	    = 2
            

  • Look at number 50.
  • Look at 52
  • Look at 56

Homework

• 1-30 every other odd (1, 5, 7, 9 ...)
• 31-39 odds
• 47- 53 odds
• 55 a.