13 Questions on Discrete Mathematics with Answers - Exam 1 | MATH 301, Exams of Discrete Mathematics

Download 13 Questions on Discrete Mathematics with Answers - Exam 1 | MATH 301 and more Exams Discrete Mathematics in PDF only on Docsity! Mathematics 301 Exam No. 1 ∗ Answers ∗ 1. Let p represent the statement “I love cheeseburgers.” Let q represent the statement “I hate French fries.” (a) In ordinary language, write down the statement (∼ p) ∧ q . ANSWER “ I do not love cheeseburgers and I hate French fries.” Note that “I hate cheeseburgers” is NOT the negation of “I love cheeseburgers,” unless you are extremely emotional about food. For example, I don’t love green beans but I don’t hate them either. On the other hand, I hate beets and love chocolate. (b) In symbols, write down the statement “If I love cheeseburgers, then I don’t hate French fries.” ANSWER p ⇒∼ q or variations. For example you can write “If p , then q ” or use the definition of implication in terms of simpler connectives to write ∼ p ∨ ∼ q . 2. Use a truth table to investigate the two statements p ∧ (q ∨ r) and (p ∧ q) ∨ r . Are these statements logically equivalent? ANSWER p q r q ∨ r p ∧ (q ∨ r) p ∧ q) (p ∧ q) ∨ r T T T T T T T T T F T T T T T F T T T F T T F F F F F F F T T T F F T F T F T F F F F F T T F F T F F F F F F F Mathematics 301 Spring 2003 Exam No. 1 2 3. Which of the following are statements? Of those that are statements, which are true and which are false? ANSWER (a) Yahoo! This is an interjection, not a declarative sentence, so this is not a statement. (b) 2 + 2 = 789 This is a statement, obviously False. (c) Don’t mess with me, man. This is a command, not a statement. (d) All statements are either true or false. This is a statement, which is also the definition of the word “statement.” Since it is the correct definition, it is true. (e) This sentence contains eight words and a period. Self-referential, but clearly true by observation. (f) This sentence is a statement. This one was a little tricky. Most of you said that it is a statement, and that it is true. That answer got full credit. Some of you said that it is not a statement; because of the self-referential nature of the sentence, I gave full credit for that answer also. It surely isn’t a false statement, however. If it is a statement at all, then it must be true, since its own statementhood is the assertion of the sentence. 4. You are given the statement “If not enough tickets are sold, or if one of the performers gets sick, then the concert will be canceled.” (a) Suppose that the concert is held as scheduled. What can you conclude about the number of tickets sold? What can you conclude about the health of the performers? ANSWER Enough tickets were sold, and the performers all stayed healthy. Be careful on the last one; remember that the negation of “one of the performers is sick” is “all the performers are healthy.” Unlike loving and hating, I think we can agree that “not sick” is the same as “healthy.” (b) Let p be the statement “The concert will go on as scheduled.” Let q be the statement “Enough tickets are sold.” Let r be the statement “The performers stay healthy.” Write down the statement at the beginning of the problem in symbolic form. ANSWER (∼ q) ∨ (∼ r) ⇒ ∼ p is the way the statement is given in the first line of the problem. However, there are many negations in this version; it might be easier to write the contrapositive and say p ⇒ q ∧ r . Mathematics 301 Spring 2003 Exam No. 1 5 9. Let P (x) be the statement “ x is a rational number.” Let Q(x) be the statement “ x2 is a rational number.” The domain of both statements (or predicates) is the set of real numbers. (a) Using logical symbols, and the letters P (x) and Q(x) , write down the statement “For all real numbers x , if x2 is rational, then x is rational.” ANSWER ∀x ∈ R , Q(x) ⇒ P (x) (b) What is the converse of the statement in part (a)? ANSWER ∀x ∈ R , P (x) ⇒ Q(x) , or “For all real numbers x , if x is rational, then x2 is rational.” (c) Is the original statement true? Is its converse true? ANSWER The converse is true. The original statement is false; for instance, if x = √ 2 , then x2 = 2 . 10. Use a truth table to decide whether the following argument form is valid: p ∨ q p ⇒ r q ⇒ r ∴ r ANSWER p q r p ∨ q p ⇒ r q ⇒ r T T T T T T ∗ T T F T F F T F T T T T ∗ T F F T F T F T T T T T ∗ F T F T T F F F T F T T F F F F T T The asterisks mark the critical rows. Since the conclusion r is true in each of those rows, the argument form is valid. As many of you noted, this form is called Dilemma or Division into cases. Mathematics 301 Spring 2003 Exam No. 1 6 11. (a) Write down the number 34510 in base 2 . ANSWER 345 = 256 + 64 + 16 + 8 + 1 = 28 + 26 + 24 + 23 + 1 , which is 1010110012 . (b) Complete the following sentence: ANSWER Suppose that x is a positive integer; then 4 is a factor of x if, and only if, the base-two numeral for x ENDS IN 00 . So many people got this wrong that I threw out this part; I obviously didn’t make it clear what I wanted. 12. You are given these statements: A. If students drink a lot then the bars will stay open late. B. If there are no unfortunate incidents, then the university will have a good reputation. C. If students do not drink a lot, then there will be no unfortunate incidents. D. If students do not pay attention to their studies, then the university will not have a good reputation. Only one of the following is a valid conclusion from the statements above. Which one? Explain your answer. Assume that the negation of “The bars stay open late” is “The bars close early.” 1. If the university has a good reputation then the students do not drink a lot. 2. If the students drink a lot then they will not pay attention to their studies. 3. If bars close early then students will pay attention to their studies. 4. If the university has a good reputation then the bars close early. Mathematics 301 Spring 2003 Exam No. 1 7 ANSWER I used the following notation: D : The students drink a lot. L : The bars stay open late. U : There are unfortunate incidents. R : The university has a good reputation. P : The students pay attention to their studies. Then the given statements are: A. D ⇒ L B. ∼ U ⇒ R C. ∼ D ⇒∼ U D. ∼ P ⇒∼ R . Following the implications—and using contrapositives when necessary—we get ∼ L ⇒∼ D ⇒∼ U ⇒ R ⇒ P . Thus, if the bars close early (do not stay open late) then the students will pay attention to their studies, which is conclusion (c). All the other listed conclusions require that you assume at least one converse of the given statements.