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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 x 2 y , determine the truth value of each of the following (a) P (1) True (b) G (2 ,- 3) True (c) P (- 1) G (- 2 , 1) True 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 ) False (b) ( xP ( x )) ( xG ( x, 0)) True (c) yG (2 ,y ) True 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 ) Solution: P (0) P (1) P (2) P (3) (b) x ( x 6 = 3 P ( x )) Solution: P (0) P (1) P (2) 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 )). Solution: Some person is a student and lives in Japan (b) x (( L ( x ) S ( x )) J ( x )). Solution: Every person who lives in Japan and is not a student speaks Japanese (c) x ( S ( x ) L ( x ) J ( x )). Solution: There is a student who lives in Japan and who does not speak Japanese (d) There is a Japanese speaking student. Solution: x ( J ( x ) S ( x ) (e) Not all speakers of Japanese live in Japan. Solution: x ( J ( x ) L ( x ) (f) Some students live in Japan, but some dont. Solution: x ( S ( x ) J ( x )) x ( S ( x ) L ( x )) (g) The only Japanese residents who dont speak Japanese are students. Solution: x ( L ( x ) J ( x ) S ( x )) 5. Determine whether each of the following pairs of sentences are equivalent. If so, explain why. If not, give an5....
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This note was uploaded on 02/06/2012 for the course MATHEMATIC 55 taught by Professor Robbayer during the Summer '09 term at University of California, Berkeley.
- Summer '09