1873_solutions

Q and r is true we could begin by writing suppose that

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online