w2 - Math 55 Worksheet Adapted from worksheets by Rob...

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Unformatted text preview: Math 55 Worksheet Adapted from worksheets by Rob Bayer, Summer 2009. Instructions Introduce yourselves! Despite popular belief, math is in fact a team sport! Decide who will be your first scribe. Do the problems below, having a different person be the scribe for each one. Try to work out the problems as a group, but feel free to flag me down if you run into a wall. Predicates and Quantifiers 1. If P (x) is the statement "x > 0" and G(x, y) is the statement x2 y, determine the truth value of each of the following (a) P (1) (b) G(2, -3) (c) P (-1) G(-2, 1) 2. Using the same predicates as above, determine the truth values of each of the following statements if the domain is the set of all real numbers. (a) xP (x) (b) (xP (x)) (xG(x, 0)) (c) yG(2, y) 3. Suppose the domain of P (x) consists of the integers 0,1,2,3. Re-write each of the following statements without using quantifiers: (a) xP (x) (b) x (x = 3 P (x)) 4. Let S(x) be the statement "x is a student," L(x) be "x lives in Japan" J(x) be "x speaks Japanese." Translate each of the following into English or into logic symbols as appropriate. The domain is the set of all people. (a) x (L(x) S(x)). (e) Not all speakers of Japanese live in Japan. (f) Some students live in Japan, but some don't. (g) The only Japanese residents who don't speak Japanese are students. (b) x((L(x) S(x)) J(x)). (d) There is a Japanese speaking student. (c) x(S(x) L(x) J(x)). 5. Determine whether each of the following pairs of sentences are equivalent. If so, explain why. If not, give an example of predicates and domains where they differ. (a) x(P (x) Q(x)); xP (x) xQ(x). (c) x(P (x) Q(x)); xP (x) xQ(x). (b) x(P (x) Q(x)); xP (x) xQ(x). Nested Quantifiers and Restricted Domains 1. Let T (x, y) be "x is taking y", L(x, y) be "x likes y", R(x, y) be "x is required to take y." If the domain for x is all students and they domain for y is all classes, translate each of the following between English and Logic: (a) xyL(x, y). (d) xy(T (x, y) L(x, y)). (b) yxL(x, y). (c) yxR(x, y). (e) Every student is required to take at least one class. (f) Some student likes all of his/her current classes. (g) Some student only likes courses which he/she isn't required to take. 2. Determine the truth value of each of the following statements. The domain is the set of all real numbers (a) xy(x > y). (c) xy(x = y 2 ). (b) xy(x y y x). (e) xy(x + y = 1 x - y = 3). (d) xyz(x > y x > z > y). (f) > 0 > 0 x (|x - 3| < |x2 - 9| < ). 3. Re-write each of the following so that negation symbols appear only after predicates. (a) xyP (x, y). (c) xy(P (x, y) Q(x, y)). (b) xP (x). (d) ((xP (x)) yz(R(y, z) S(y, z))). 4. The symbol !xP (x) stands for "there exists one and only one x such that P(x) is true" and is often pronounced "there is a unique x..." Show that !xP (x) can be rewritten using just regular quantifiers. Puzzles 1. Consider the following set of four statements: (a) One of these statements is false (b) Two of these are false (c) Three of these are false (d) Four of these are false Which of the above, if any, are true? 2. Two bicyclists enter opposite ends of a 100-foot long tunnel that is only wide enough for one bike. One is travelling 10 ft/s and the other is travelling 5 ft/s in the opposite direction. A bird flying 20 ft/s enters the tunnel just in front of the 10 ft/s cyclist. When the bird gets to the other cyclist, it immediately turns around and flies back towards the 10 ft/s cyclist. When the bird gets back to him, it turns around again, etc, etc. How many feet does the bird fly before the bicyclists collide? 3. Four people travelling at night come to a small footbridge and need to cross to the other side. Unfortunately, they only have 1 flashlight and the bridge can only hold the weight of two people at once. One person takes 10 minutes to cross, another 5, another 2, and the last takes 1 minute. Anyone crossing must have the flashlight and when travelling together, they must go the pace of the slower person. What is the quickest time in which that can all get across? ...
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