MAT 101 Exam 2 Logic (Part I) Fall Circle the correct answer on the following multiple-choice questions.

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1 Name: MA 101 Exam 2 Logic (Part I) all 2017 Multiple-Choice Questions [5 pts each] Circle the correct answer on the following multiple-choice questions. 1. Which of the following is not a statement? a) All prime numbers are odd naturals greater than 1. b) is larger than 10. c) riangles do not have 4 sides if and only if squares have exactly 5 corners. d) Sentences consisting of ten words or less are always false. e) or some integer x, x x If p is true, then what is the truth value of the compound statement ~ p (~ q r) a) rue. b) alse. c) his truth value cannot be determined.? 3. What can you conclude, in general, about the symbolic statement ~ pq, are component statements? a) It is a tautology (i.e. it is always true). b) It is a contradiction (i.e. it is always false). c) It is only true when q is true. d) It is only true when both p and q are true. p p q, where 4. How many rows would be needed to construct the truth table of a compound statement with 6 component statements? a) 12 b) 16 c) 32 d) 64

2 or questions 5 7, let w represent the statement he wife has long hair, h represent the statement he husband has a beard, and r represent the statement I remember this couple. 5. Which of the following is the symbolic representation of the statement It is not the case that the wife has long hair or that the husband has a beard. a) ~ w h b) ~ w ~ h c) ~ w h d) ~ w h 6. Which of the following is the verbal translation of the symbolic statement (~ w) (~ h)? a) Neither does the wife have long hair nor does the husband have a beard. b) he wife does not have long hair or the husband does not have a beard. c) It is not the case that the wife has long hair or the husband has a beard. d) Either the wife has long hair or the husband has a beard. 7. Which of the following is equivalent to the symbolic statement ~ w h r? a) he wife has long hair and the husband does not have a beard, provided I remember this couple. b) I remember this couple, provided the wife has long hair and the husband does not have a beard. c) I do not remember this couple, unless the wife has long hair and the husband does not have a beard. d) If I remember this couple, the wife has long hair and the husband does not have a beard.

3 Show all your work on the following problems to receive full credit. Problem 1 [15 pts] Write the following song lyrics symbolically. Specify the letters you attribute to each of the component statements in the quotation explicitly. a) You may say I m a dreamer but I am not the only one. - John Lennon b) Ready or not, here I come. - he Delfonics c) Lean on me when you re not strong and I ll be your friend. - Bill Withers

4 Problem 2 [15 pts] Write the negation of the following statements. Do not just provide symbolic answers. 11 a) b) She was unwilling to join the party or her car broke down that day. c) Everyone on Long Island was affected by super storm Sandy in 2012, but some did not lose power in their homes.

5 Problem 3 [15 pts] a) Use the truth tables provided below to show that ~ ~ p q is not logically equivalent to p ~ q, where p and q represent any two component statements. p q p q b) Can you prove that ~ ~ p q p ~ q without using truth tables? [Bonus 5pts]

6 Problem 4 [20 pts] a) Using the truth table below, show that the symbolic statement (~ p q) (~ q r) is a tautology (i.e. a compound statement that is always true.) p q r b) Consider the following three component statements: p All triangles have more than 3 sides. q Odd integers have a factor of 2. r Some pigs can solve differential equations. Using the truth table constructed in part a), how can you justify that the statement (~ p q) (~ q r) is still true with the components given above?

7 Bonus Problem [5 pts] In propositional logic, the disjunction p or q (written symbolically as p q) is said to be inclusive since it is true when either p is true, q is true, or both p and q are true. On the other hand, the disjunctive form Either p or q is said to be exclusive since it is not true when both p and q are true. Write this exclusive form of the disjunction symbolically along with its truth table.