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CS 173: Midterm Exam 1
Spring 2005
Name: SOLUTION
NetID:
Lecture Section:
General Directions
- Make sure your name is on every page.
- There are 11 pages, including a sheet of scratch paper. Make sure that you answer all 14 questions.
- Remember to write clearly and legibly. Unreadable answers will receive no credit.
- This is a closed book exam. No notes of any kind are allowed.
- Remember to time yourself.
Question Points^ Out of 1 5 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 10 10 10 11 10 12 10 13 10 14 10 Total 100
Multiple Choice
Problem 1 (5pts)
Which one of the following functions is a bijection from ^ to ^?
a) b) c) ^ ^ ! " d) (^) ##%$
Solution
&
Problem 2 (5pts)
Which one of the following propositions is NOT a tautology?
a) ('),+-). ' 0/ b) 3'4,+5 /63'7),. c) 9.+:);(',/<+5 = />. ' d) 3'/>+5 ?)@+A/215 B/C3'/
Solution
D
Problem 3 (5pts)
Let EF HGIKJLGMI?NOJLGMIPJQGIPN NRN .Which one of the following statements is true?
a) GMIKJLG GMIPN N N is in the power set of E. b) EHS;GMI?NT I. c) Let U HGMIPJQGIPN N. Then UWV,E and UWXYE. d) E[Z\GRGIPN N] HGIKJLGMI?NOJLGRGIPN NOJLGIKJQGIPN NRN
Solution
&
Problem 7 (5pts)
Given the predicate symbols
x is “ is a dog.” y is “ is a rabbit.”
^zKJ{| (^) is “ chases {.”
Which one of the following is an appropriate translation of the English statement “Only dogs chase rabbits”?
a) ~} {| Lm Q y {| ?)^zPJ={| B/ x = b) 9m{| L~} L y {| ?)^zKJ{| B/ x c) } {| Q} L= y {| )^zKJ{| B/ x d) 9m{| Lm Q y {| P)^zKJ{| B/ x
Solution
c
Problem 8 (5pts)
Which one of the following arguments is NOT valid?
a) If is a real number with @ f , then F. Suppose that v . Then . b) If is a real number with @ , then Kv . Suppose that . Then P . c) All students in this class love CS173. Xavier is a student in this class. Therefore, Xavier loves CS173. d) Every computer science major takes CS173. Natasha does not take CS173. Therefore, Natasha is not a computer science major.
Solution
b
Short Answer Problems
Problem 9 (10pts)
Prove the inference rule known as ”simplification” without using truth tables.
Solution
'),+F ?'F.w3'),+5 ?4c'e. 'c4.+5 ?4c'e3'74,. ' 4,+vq
Comments
Each mistake drops two points.
Problem 11 (10pts)
Prove or disprove the following statements.
a) For any integers l and j, lz A FjHz A/<lz , Fjz o. b) For any integers l and j, l 7 oA jLz />lz oA jHz o.
Solution
a) We first prove that l7 "1A je7 oA/<l z ov Fj 7 o.
Let l7 F v d1 , and j: FQT d1 , with 1 cVG ¡JQ J%¢£¢ ¢ J|N. (Any integer can be written in this form.) Then l R % 515A [1 , and j RQ " 51L] [1 . Thus, lmk ¡9RB f515| P [1T Y1M¤7 o. Similarly, jLk Y1M¤z m. b) We disprove that l z ov Fj 7 oA/<lz , FjHz ,.
All that is needed is a counterexample. Let lz "¥ , jw . Then, l| z mo j z ". However, l\¦j.
Comments
a) 5 points. 1 point for lz A [. 1 point for j: %] [. 1 point for squaring. 1 point for factoring. 1 point for the argument. b) 5 points. 1 point for recognizing that the claim is false. 4 points for the counterexample.
Problem 12 (10pts)
Prove that if _fXq , then _oZaq" q.
Solution
The proof can be done directly or indirectly, but similar intricacies exist in each. Assume Xq. First, show ZqFXq. Choose oVoZq. Then, \Vo or oVq by definition of union. Case 1, \V_. Then oVq since _[Xq. Case 2, oVq.
Second, show q§XY_Zq. Choose oV,q. Then oV_,Zq by addition.
Comments
10 points. 2 points for a reasonable attempt. 1 point each for writing down correct definitions of X , and Z. 1 point for assuming _fXq. 2 points each for _oZaq§Xq , and vice versa. 1 point for cases.
Solution
a) R b) c) ¨ZqbZ,E ¨ ¨ ¨ ¨q ¨ ¨ E
_
oZaq ¨ ¨q[ZE ¨ ¨,Z,E ¨ ¨qb,E ¨ © % R w % R R kdTbTd© R% RªR
Comments
a) 3 points This one is saying that set _ (students that are sick) is a subset of q and q is a subset of E. So _Z7q;Z!E is just E. Therefore,
_ZqdZE
E
.
b) 3 points Pairwise disjoint means the every pair of sets does not have overlapping part. Therefore,
_Zdq§ZfE
_
q
E
.
c) 4 points This is a standard problem of set operations. First, we add up ¨_ ¨, ¨q ¨, and ¨E ¨. Since the green parts (i.e. ¨_`q ¨, ¨qoE ¨, and ¨_;oE ¨) are added twice
and the red part (¨ ¨_,q[,E ¨) is added three times, we next subtract duplicate ones. After subtracting ¨oq ¨, qfoE ¨, and ¨;oE ¨from the sum, we find that ¨_;oqfoE ¨is actually subtracted three times. So we finally add ¨_qdE ¨once. Therefore, the overall process looks like: ¨_ZqbZ,E ¨ ¨_ ¨ ¨q ¨ ¨E ¨ ¨_Zq ¨ ¨q[Z,E ¨ ¨_oZE ¨ ¨_qd`,E ¨
Problem 14 (10pts)
Let «! , ¬! , and y be the statements ” is a duck,” ” is annoying,” and ” is a dancer,” respectively. Express each of these statements using quantifiers, logical connectives, and «! , ¬ , and y .
a) All ducks are annoying. b) Some dancers are not annoying. c) Some dancers are not ducks. d) Does c) follow from a) and b)? Prove your response.
