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Unformatted text preview: CSE 260002 Fall 2010 Examl ANSWER October 1 Name: This is a closed—book exam, with 6 pages and 11 problems totaling 100 points.
A reference sheet is also provided with known logical equivalences. Time 50 minutes
Logic 1. (10 pts) Determine the truth value of the following statements.
Circle either True, False, or Undetermined. *T F U If 2 + 3 7E 5, then you will get an A in this class.
*T F U (30 /\ (p —> (1)) —> q is a tautology T F *U If it is raining it is cloudy. *T F U (30 V org) /\ (g V 7") logically implies p V 7" T *F U q —> p is the contrapositive of p —> g. 2. Consider the following proposition P:
P: “If it is raining it is cloudy and humid.”
Propositions r, c, h making up P are as follows:
0 r: it is raining
o c: it is cloudy o h: it is humid (a) (6 pts) Give the well formed formula for P using r, c, h, and logical connectives.
r —> (c /\ h) (b) (6 pts) Give the following if it can be determined based on the above ONLY.
If it can’t be determined, write “Undetermined.” i. a necessary condition for raining.
c /\ h ii. a necessary condition for humid.
undtermined iii. a suﬂicient condition for cloudy.
7" (c) (4 pts) Give the contrapositive for the well formed formula P in part (a) and then
express the contrapositive in plain English. ﬁ(c/\h) —> ﬁr If it is not cloudy or humid then it is not raining. Logical Equivalences l. (5 pts) Using a truth table, determine whether the following is a tautology, contradic—
tion or a contingency. (n61 M10 e (1)) e 10
After giving the truth table, do NOT forget to state and justify your answer! 29 q p—HI a]? ﬁg (MAW—>90) (cg/\(P—wD—HP T T  T  F  F  F  T
T F  F  F  T  F  T
Fl T  T  T  F  F  T
Fl F  T  T  T  T  T
rTautology 2. (8 pts) Using substitution of logical equivalences ONLY (given in the table on the last
page of this exam), show the following logical equivalence.
Do NOT forget to justify each step with a rule! p—Nq/Wﬁiﬁpew/WP—W) p —> (q /\ T) 1% ﬁp V (q /\ 7") implication law
1% (ﬁp V q) /\ (ﬁp V r) distributive law
(if (p a q) /\ (p a 7") implication law Predicates and Quantiﬁers 1. (10 points) Consider the following statements: R(X): X is a rich person
F(X): X is a famous person
H(X): X is a happy person universe of discourse for the variable x is persons in the world. Write logical expressions using predicates? quantiﬁers, and logical connectives for the
following. (a) If any rich person is famous then the person is also happy.
WHEEL") /\ FWD e HM) (b) Some happy persons are rich but not famous. EIAHL’E) /\ R(:c) /\ off—Tm» (5 points) Give the truth values of S in the following table and then write in plain
English the meaning of the statement 8. Assume the universe of discourse is {1, 2, 3} S: Elxﬁoﬁc) /\ Vyﬂl“ 75 y) —> 10(9)) 2. p(1) p(2) p(3) S
T F F T
F T T F S is true when p is true for exactly one value of X. 3. (8 points) Let Student(x,y) be deﬁned as follows:
Student(x,y): Student X is taking course y Consider the following predicate logic statement: EIyEIAStudenﬁx, y) /\ Student(z, y) /\ z : ”John” /\ x 7E ”John”) Universe of discourse for x and y are all students and all courses in the university,
respectively. Answer the following: (a) What are the bound and free variables for the above? bound: y,Z
free: x (b) What are the values of the free variable that make the above true?
X is a student taking at least one class that John is taking. Sets and Set Operators 1. (14 points) Let A = { {a}, {av {b}}}. Determine whether each assertion in (a)i(g) is
true or false. (mngmm
(b) (D E A false
(0) {a} g A false
(d) {a} e A true
(6)
(f) {b} e A false
) {b} E A false (g {m} g {a} true 2. Let A: {1,2 ? 7 ? 7 3 4 5 7},B= {1,3,8,9},C= {1,3,6,8} (a) (7' points) Draw a Venn diagram representing A, B and C. (b) (6 points) Answer the following as true or false, by referring to the Venn diagram
above. (AﬂBﬂC‘) true
0 36((A—B)UC) true
[(AUB) *0) false B AUB AUB A B AﬂB OOHH
OHOH
QHHH
I—LOOO
I—LI—LOO
HQHO
HOOD Fourth and seventh columns are the same....
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