Inductive Reasoning

• Inductive Reasoning: The process of reasoning to a general conclusion through observation of specific cases.
• Example:
  • What can we say about the sum of two odd numbers?
  • 1 + 3 = 4 ✓
  • 7 + 3 = 10 ✓
  • 1001 + 33 = 1034 ✓
  • So we decide The sum of any two odd numbers is even
  • Notice, we look at some special cases and
• This is opposed to a deductive reasoning
• Deductive Reasoning is the process of reasoning to a specific conclusion from a series of general statements.
• Example:
  • Prove the sum of two odd numbers is an even number.
  • let a and b be odd numbers.
  • a = 2n + 1 and b = 2m+1, the definition of odd and even
  • a+b = 2n + 1 + 2m + 1, the definition of sum.
  •     = 2n + 2m + 2
  •     = 2(n + m + 1)
  • but 2(n + m + 1) is even.
• Problems:
  • Inductive reasoning can lead to incorrect results:
    • It snowed yesterday
    • It snowed today.
    • Therefore, it will snow tomorrow
  • Our conclusion might be wrong
  • We might have observed insufficient data to form a proper conclusion
  • There might not actually be a pattern where we are looking for one.
  • Always look for a counter example.
• In general, what we are looking for is a pattern.
• Examples
  • Problem number 12, page 6

               Use inductive reasoning to predict the pattern:
               13 x 10 = 130         13 x 11 = 143
    	   13 x 12 = 156         13 x 13 = 169
    
    	   solution:
    	      Notice that the product increases by 13 each time
    	      (130 + 13 = 144, 143+13 = 156, 156 + 13 = 169)
    
    	      13 x 14 = 169 + 13 = 182
           

  • Problem 16, page 6

               Draw the next figure
    
    	   :)  :( :|  :)  :(  :|
    
    	   The pattern seems to be
    	     smile, frown, neutral, so the next item will be neutral
           

  • Problem 21

              Predict the next number in the pattern
    
    	  1, -1, 1, -1, 1, -1, 1, ...
    
    	  The pattern seems to be multiply the previous value by -1
    
    	  so -1 is the next number
           

  • Problem 29

               Predict the 118th entry in the pattern
    	   Y, R, R, Y, R, R
    
    	   This takes a bit more work.   Notice  that there are three letters
    	   in the pattern.
    
               1/3 = 0 R 1
    	   2/3 = 0 R 2
    	   3/3 = 1 R 0
    
    	   position   1 2 3 4 5 6 7 8 9
    	              Y R R Y R R Y R R
               remainder  1 2 0 1 2 0 1 2 0 
    
    	   118/3 = 39 R 1, 
    	   There is always a Y in the R=1 position, so the answer is Y
           

  • Problem number 41.

           Pick any number and add 1 to it.  Find the sum of the new number
           and the original number.  Add 9 to the sum.  Divide the sum by 2
           and subtract the original number from the quotient.
    
           Work this out for a number  
              Pick 0   0+1 = 1
    	           1+0 = 1
    		   1 + 9 = 10
    		   10/2 = 5
    		   5 - 0 = 5
    
           Let n be the original number
           Add one to it				n+1
           Find the sum of the new number
               and the original number 		n+1+n
    	    					2n+1
           Add 9 to the sum                         2n+1+9
           						2n+10
           Divide by 2                              (2n+10)/2
                                                    n+5
           Subtract the original number		n+5-n
           						5