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FinalExam20081216 - Csci 2610 Fall 2008 Final Exam...

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Csci 2610, Fall 2008 Final Exam 2008.12.16 1 Name: _________________________________ Seat:_______________ The Final Exam has 14 questions worth a total of 100 points. Please make certain you have all 8 pages. You will have 180 minutes to complete the exam. You are allowed a two-sided sheet of paper with notes. You may use a calculator. Unless otherwise instructed, give all probabilities as decimals, percents, or fractions (e.g., 0.0568, 12.3%, or 5/6). 1. (5 points) Show that ((p r) (q r)) is logically equivalent to (p q)   r. 2. (5 points) Let P(x,y) be the statement “x is a parent of y.” (Or “y is a child of x.”) Let R(x,y) be the statement “x is related to y.” (This refers to a family relation). The universe of discourse for both predicates is all people. Using quantifiers, write the following propositions: a) Everyone has a parent. b) Any two people must be related if they are parent and child. c) Someone is a grandparent. d) A predicate Q(x,y) is symmetric iff x y(Q(x,y) Q(y,x)). Which of the above predicates is symmetric?
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Csci 2610, Fall 2008 Final Exam 2008.12.16 2 3. (5 points) Use a symbolic proof to derive x( P(x) S(x)) from the premises x(R(x) Q(x)), x(S(x) R(x)), and  x(P(x) Q(x)).
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