1.1, Inductive Reasoning

• We will start by working on problem solving skills.
• It is important to think about how we solve problems.
• Inductive reasoning : The process of reasoning to a general conclusion through observation of specific cases.
• Deductive reasoning: The process of reasoning to a specific conclusion from a series of general statements.
• Each has a place in problem solving.
• Everyone uses the first, inductive reasoning, all of the time.
• Mathematicians use the second in very special cases.
• Inductive Reasoning Example: The sum of any two even integers is even.

•    2 + 2 = 4 ✓  2 + 4 = 6 ✓  4 + 4 = 8 ✓
     

• Deductive Reasoning Example: The sum of any two even integers is even

•    Let x and y be any two even integers.
         x = 2×a, y = 2×b (the definition of even integers)
         x+y = 2×a + 2×b
             = 2×(a+b) (Distributive Property )
       let a+b = c, some other integer
         x+y = 2×c, which is even.
     

• Inductive reasoning can lead to errors:
  • If insufficient cases are observed to see the general pattern
  • "I dropped my baby on it's head and it is fine, therefore dropping babies on their heads is not a problem."
  • Or if no general pattern exists.
• But inductive reasoning must be used when we can't do deductive reasoning (see problems 35-36)
• It is often useful to write a conjecture or a statement of your general conclusion.
• If a special case is found that proves the conclusion is wrong, it is called a counterexample
• Do some problems (Starting on page 6):
  • number 12
  • number 20
  • number 32
  • number 36
  • number 40
  • number 44

Homework: