# exam1v4 - CSE 260 Fall 2011 Name/PID:— 1 1(10 points...

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Unformatted text preview: CSE 260, Fall 2011 Name/PID:— 1 1. (10 points) Assume the universe of discourse is the set of rational numbers. Indicate if the logic formula is true or false by circling the correct response. True False ﬁVu (a : a — 1) ANS: T True False VuEv (v 152) ANS: T True False VuEv (a v2) ANS: F True False :uVU (av : a) ANS: T True False ﬁVut/v (1; >1; e (u 7 v)(u + v) > O) ANS: T 2. (10 points) Let x range over all personnel of some company and 71 range over all possible security levels. Deﬁne the predicate L as follows L(a:, n): “The security—clearance level of person :1: is 71.” Translate the following statements into predicate logic formulas. (a) No one at the company has a higher clearance level than 10. ANS: ﬁ*\$*n(L(\$,n) /\ n > 10) (b) Someone has the lowest security level (ie. a security level lower than that of anyone else at the company). ANS: 55 nVyVm (L(5c,n) /\ (L(y,m) 4w n g m» CSE 260, Fall 2011 Name/PID:— 2 3. (10 points) For each of the following arguments, determine if the reasoning is valid. Additionally, justify your answer by either indicating the rule(s) of inference used, if the reasoning is valid; or showing that the conclusion is not a logical consequence of the premises, otherwise. Use the names in the handout for identifying rules of inference. a The program crashes if the user enters 8. string or the number 0 Although the program crashed the 7 user did not enter 0 Therefore, the user entered 3. string. ANS: Invalid. The form of the argument is: (8 V 2) 4» c and c /\ oz, therefore 8. But ((8 V 2) —> c) /\ (c/\ oz» 4w 8 is not a tautology. For example, if 3 and z are both false and c is true, it is false. This answer formalizes that, even if either of these events could cause a crash, other events (e.g., a power surge) could also. (b) All jobs that are submitted by a system administrator are high—priority jobs. Adam Pitcher, who is a CSE system administrator, has submitted a job. Therefore, there is a high—priority job. ANS: lts valid. The inference rules used are: Universal instantiation, existential instantiation, modus ponens, and existential generalization. 4. (5 points) Rewrite the following formula so that negations apply only to predicates (that is, so that no negation is outside a quantiﬁer or an expression involving logical connectives). says/a: (Qty) a We. y) v QM) ANS: was (my) A Ray) A was» CSE 260, Fall 2011 Name/PID:— 3 5. (12 points) Let A = {{0 }, {0, 1 }, {0, 2}} and let 73(8) denote the power set of S, for any set S. For each logic formula below, indicate if it is true or false by circling the correct response. True False (ll 6 A ANS: F True False (D C A ANS: T True False {O } E A ANS: T True False {O} C A ANS: F True False 1 g?! A ANS: T True False A C ?({0,1, 27374” ANS: T 6. (6 points) Show that the following propositions are not logically equivalent, where the universe of discourse is 73(R), the set of all sets of real numbers. (1) VSVTVV ((3 E V) /\ (T E V» (ii) VSVTVV ((3 (1T) g V) ANS: They are not logically equivalent since it is possible for (ii) to be true and for (i) to be false. For example, if S is the set of real numbers and if T and V are both the set of integers, then (ii) becomes (R H Z) g Z, which is true, and (i) becomes (R Q Z) /\ (Z Q Z), which is false. CSE 260, Fall 2011 Name/PID:— 4 7. (10 points) Prove that (op E; g) /\ (r E; p) and 10V ((1 Am") are logically equivalent using the derivation method, in which you derive one of the formulas from the other. For each step in your derivation, indicate the equivalences that you used to justify the step. (Refer to the sheet for the names of the equivalences. You may abbreviate names by leaving out ”Law” and by shortening long names to the ﬁrst 5 letters.) ANS: (ﬁp —> g) /\ (T —> p) E (p V q) /\ (ﬁr V p) Conseq. (twice) 85 double neg. E (p /\ (or Vp)) V (q /\ (or V p)) Distributive Law E (p /\ or) V (p /\ p) V (g /\ —.7~) V (g /\ p) Distributive (twice), associative E [(p /\ or) Vpl V (q /\ or) V (q /\ p) ldempotent, assoc. E p V (q /\ ﬁr) V (p /\ 9) Absorption E (p V (p /\ (3)) V (q /\ W") Commutative, assoc. 10 V (9‘ A ﬂT) Absorption ...
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