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Unformatted text preview: w that R
must be true.
b. Write down the assertion P Q
this form. R in its contrapositive form and outline a strategy for proving it in The contrapositive form of the assertion P Q R says that
Q
R P
To prove the assertion this way we should assume that both Q and R are false and then use
this information to show that P is false.
3. a. Outline a strategy for proving an assertion that has the form P Q R. Solution: Assume that both of the statements P and Q are true and write a proof that R is true.
b. Write down the assertion P
this form. Q R in its contrapositive form and outline a strategy for proving it in Solution: The contrapositive form says that R P
Q . Assume that the statement R is
false and show that at least one of the statements P and Q must be false. 4. a. Outline a strategy for proving an assertion that has the form P Q R. We need to write two proofs. First we need to show that if we assume P then R must be true.
Then we need to show that if we assume Q then R must be true.
b. Write down the assertion P
this form. Q R in its contrapositive form and outline a strategy for proving it in The contrapositive form of the assertion P Q R says that R P
strategy for proving this form of the assertion was given in Exercise 1a.
5. a. Outline a strategy for proving an assertion that has the form P QP Q and the R. Solution: Assume that the statement P is true and write a proof that R must be true. Then
assume that the statement Q is false and write a proof that R must be true.
b. Write down the assertion P
proving it in this form. QP R in its contrapositive form and outline a strategy for Solution: The contrapositive form says that R P
Q. Assume that R is false and write a
proof that P must be false. Then assume that R is false and write a proof that Q must be true. Exercises on Statements Containing Quantifiers
1. Physicist’s proof that all odd natural numbers are prime: 1 is prime. 3 is prime. 5 is prime. 7 is prime. 9 is
experimental error. 11 is prime. 13 is prime. We have now taken sufficiently many readings to verify the
hypothesis. Comment!
2. You know that there are 1000 people in a hall. Upon inspection you determine that 999 of these people are
men. What can you conclude about the 1000’th person? Solution: You can’t make any conclusion at all about her. Don’t even try.
12 3. The product rule for differentiation says that for every number x and all functions f and g that are
differentiable at x, we have
fg x f x g x f x g x .
Write down the opening line of a proof of the product rule. Your opening line should start: Suppose that ... Solution: Suppose that x is a real number and that f and g are functions that are differentiable at the
number x.
4. Given that P x and Q x are statements that contain an unknown x and that S is a set, outline a strategy for
the proving the assertion P x Q x for every x S. Write down the opening line of your proof. Solution: Suppose that x S and that the condition P x is...
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 Fall '08
 STAFF
 Math, Calculus

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