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Computer Science 70 - Spring 2002 - Vazirani - Midterm 1

# Computer Science 70 - Spring 2002 - Vazirani - Midterm 1 -...

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CS 70 file:///C:/Documents%20and%20Settings/Jason%20Raftery/Desktop/R... 1 of 5 5/12/2006 8:44 PM CS 70 Discrete Mathematics for CS Spring 2002 Vazirani Midterm 1 PRINT your name: ____________________________SID________________________ This is a CLOSED BOOK examination. Do all your work on the pages of this examination. Give reasons for all your answers. 1. (15 pts.) Satisfiability and all that For each of the following Boolean expressions, decide if it is (i) valid (ii) satisfiable (iii) unsatisfiable. (Give all applicable properties.) (a) (10) [(P ===> Q) ^ (Q ===> R)] ===> (P ===> R) (b) (5) ~(P ===> Q) ^ ~(Q ===> P) CS 70, Spring 2002 Midterm 1 2. (Each 10 pts.) Logic and Proofs (a) Can you define open sentences (i.e., sentences whose truth value depends on some variable x) P(x) and Q(x) and a universe U so that (for all x in U)(P(x) ==> Q(x)) is false, and (for all x in U)(Q(x) ==> P(x)) is false?

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CS 70 file:///C:/Documents%20and%20Settings/Jason%20Raftery/Desktop/R... 2 of 5 5/12/2006 8:44 PM If yes, give an example. If no, explain why not. (b) Write a DNF formula that expresses the constraint that at least two of X1, X2, X3, X4 are true.
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Computer Science 70 - Spring 2002 - Vazirani - Midterm 1 -...

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