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NOTE 2: When two or more Lessons are to be turned in on the same day, please staple the Lessons separately and turn them in separately.
(This is Lesson 1a; if another Lesson is due the same day as this Lesson is due, staple the two Lessons separately and turn them in separately.)

Due Friday, September 5

Last modified Sep 4, 2008 1:23 PM

• Regarding Exercise 8 in Section 1.1:
  You were asked in Exercise 8 to use the Principle of Mathematical Induction. State clearly and carefully what the statements  P(m)  are that you are proving by induction. Be sure to introduce all variables appropriately.
  
  
• Regarding Exercise 7 in Section 1.1:

  Exercise 7 asks you to prove a certain basic fact about the natural numbers, and it gives you a hint to help you do this, the hint suggesting that you prove that a certain set is an inductive set.
  
  Several important points here:

  • First, if you're going to use the hint and prove that the given set is an inductive set, YOU HAVE TO KNOW WHAT THE DEFINITION OF “INDUCTIVE SET” IS.
  • Second, being asked to prove that a certain set is an inductive set (compare the first point) is not the same as being asked to prove that a certain statement is true by using induction (although in certain cases you may want to use induction to do your proof).
  • Third, if you ARE going to prove something by induction, it has to be clear to you and your reader what the statement is that you are proving by induction (although in Exercise 7, induction is not required).
  • Fourth, this exercise is given at the beginning of the development of the properties of integers and natural numbers, using the definition of natural numbers given in the textbook. Thus, to do your proofs at this stage, you can't use properties of the natural numbers which will be proved later. Basically all you know now, in addition to the definition, is that sums and products of natural numbers are again natural numbers. It has NOT yet been proved that, e.g.,
        if n is a natural number and n > 1, then n ≥ 2,
    or even
        if n is a natural number, then n ≥ 1
    (cf. Proposition 1.6 in Section 1.2, later in the book).
  • Fifth, the point of the exercise is not simply to prove the hint. The point is to prove the fact which the hint is supposed to help you to prove. Proving the hint is not the end of the problem; you still have to USE the hint to prove the main, original fact you're asked to prove.
  With this background, do the problems given here.

Last modified Sep 4, 2008 1:23 PM

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