Lesson 16b.

NOTE 1: Be sure to read Homework Format and Homework Writing Policy BEFORE writing up your solutions to be turned in.

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.

Due Wednesday, February 27

Sect. 4:

Below is a quote from Lesson 16.

Sect. 4: 4.13*(c)** * Be efficient in your proofs (e.g., don't repeat work that you've already done). Note that usually the best way of proving something about IRRATIONAL numbers is to use some kind of negation or contrapositive or contradiction argument, since the definition of IRRATIONAL is simply NOT RATIONAL. **Follow the hint in the book for 4.13(c), making sure you know what the book means by a “proof by contradiction”.

Part I

To make sure you know that the book means by a “proof by contradiction”, LOOK IT UP and then answer the following:

1. Explain how you would give a “proof by contradiction” of a statement of the form
         IF p, THEN q.
2. Explain how you would give a “proof by contradiction” of a statement of the form
         IF (p AND q), THEN r.
3. Explain how you would give a “proof by contradiction” of a statement of the form
         IF p(x), THEN q(x).
4. Explain how you would give a “proof by contradiction” of a statement of the form
         IF (p(x) AND q(x)), THEN r(x).
5. Explain how you would give a “proof by contradiction” of a statement of the form
         For all  x,  IF p(x), THEN q(x).
6. Explain how you would give a “proof by contradiction” of a statement of the form
         For all  x,  IF (p(x) AND q(x)), THEN r(x).

Part II

(Independent of Part I)
There is another way of doing Exercise 4.13(c) which does not involve a proof by contradiction (contradiction is the way suggested in Lesson 16). Prove the result of 4.13(c) as follows:

• Take the contrapositive.
• Use a form of the equivalence discussed in the extra problem in Lesson 5. (Remember that you want to try to avoid dealing with irrational numbers.)
• Use Exercise 4.13(a) and, very slightly, Exercise 4.13(b).

NOTE: As stated on the Course Home Page, all due dates are tentative. Assignments, or parts of assignments, may be postponed to a later date.

Last modified Feb 19, 2008 6:18 PM

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