math55-fall05-mt1-Bergman-soln - George M. Bergman Fall...

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Unformatted text preview: George M. Bergman Fall 2005, Math 55 19 Sept, 2005 100 Lewis Hall Solutions to the First Midterm 3:10—4:00 1. l30 points, 10 points each.) Short answer questions. A correct answer will get full credit whether or not work is shown. An incorrect answer may get partiai credit if work is given that follows a basically correct method. (a) Give the truth table for the proposition (q+~> p) A p. (Your table must show columns for p, for q and for this compound propositiou. It may or may not have other columns.) Answer: (The column q——) p is not required, but is useful in computing the last column.) p q q~>p (carom/w (b) Let f: R ~> R be given by the rule f(x) = 3x+ 1, and g: R —-> R by the rule g(x) = 4x+ 1. Then f°g: R ——> R is given by the rule Answer: (fog)(x) = 12x+4. (c) Write in mathematical symbols the statement that every real number which is not an integer lies between some integer and its successor (where the successor of an integer n means the integer n+1. If your statement is long, you don’t have to put it all on one line.) Answer: V xER—Z El nEZ (n < x < n+1). Variants are possible, for instance, V x ((xER) A r-t(xEZ)) —> (El n((nE Z) A (n < x < n+1»). 2. (24 points, 8 points each.) Complete the following definitions. Your definitions do not have to have exactly the same wording as those in the text, but for full credit they should be clear, and be equivalent to those. (a) If X and Y are sets, and f: X —> Y is a function, then the graph of f is defined to be the set . . . Answer: {(x, y) [ 105 X, y =f(x) }. (Also 0K: {(x,f(x)) | xEX}.) (b) Let I be a set, and suppose that for each ieI we are given a set Ai. Then Uiej Al» denotes . . . {For full credit, use set-builder notation rather than words.) Answer: {x E El iEI (xE A10}. (0) If f: R —) R and g: R —> R are functions, then one says that f(x) is Q(g(x)) (in words, “f(x) is big—Omega of g(x)”) if. . . Answer: El k>0 El C>0 Vx>k (|f(x)l Z Clg(x)|). 3. (30 points, 15 points each.) Short proofs. in giving the proofs asked for below, you may call upon definitions and results proved or asserted in the text. You do not have to use the formal names of methods of proof, or any standardized format, as long as your arguments are clear and logically sound. (a) Suppose X, Y and Z are sets, and f: X—> Y a function. Prove that f(X—Z) 2 f(X) -f(Z)~ Answer: To show this inclusion, we must prove that every element of f(X) ~f(Z) belongs to f(X—Z). {f aEf(X) —f(Z), that means aEf(X) but atEf(Z). Since aef(X), we can find some bEX such that a =f(b). Now if h were a member of Z, we would have f(b)Ef(Z), i.e., aEf(Z), which we have just noted is not true. Hence be Z, 50 since bEX we have bEX—Z. Since a =f(b), this says that aef(X—Z), as required. (b) Suppose that f: R —9 R and g: R —> R are functions such that f(x) is O(g(x)). Prove that f(log(|xl)) is 0(g(log(lxl))). (Recall that in this course, “log” means the logarithm to the base 2. You may use well—known facts about the logarithm function.) Answer: By the definition of the statement that f(x) is O(g(x)), there exist real constants C and k (“witnesses” to the big—0 relation) such that for all x > k one has S C |g(x)l. I claim thatfor all x > 2k one has lf(log(lxi))| S C |g(log(|x|))l. Indeed, if x > 2k then logflxl) = iog(x) > log 2" = k, so the inequaliryfor f holds with log x in the role of x, giving the asserted inequality. Hence C and 2k are witnesses to the statement that f(iog(|Xl)) is 0(g(10g(|xl))). 4. (16 points} Write in pseudocode an algorithm “intersect” which takes two sequences of real numbers a1, .5. , am and b1, , bn, where the elements of each sequence are distinct (i.e., for i at j, one has ai ¢ a- and hi it bj) and creates a sequence c1, , er of distinct real numbers such that J {(21, , cr} = {a1, , am} (1 {191, , bn}. That is, after the algorithm has run, r should equal the number of elements in that intersection, and c1, , or should be the distinct elements in that intersection. For full credit you should use only the basic operations given in the text, and follow the format for pseudocode specified there. (However, you are not expected to give any equivalent of the changes between bold italic and roman font that the text uses.) Answer: I will use fonts as in the text, though as stated, your exams are not expected to slim I anything similar. Here is one solution: procedure intersect(a1,, a : distinct real numbers, [91, , b”: distinct real numbers) m r :2 0 r for i:=1 to m begin for. j:=1 to n begin if ai : bj begin r 2: r+1 Cr 1: LI!- end end end I" Reminder: The reading for Wednesday is #7 1 ...
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This note was uploaded on 03/08/2010 for the course MATH 55 taught by Professor Strain during the Spring '08 term at University of California, Berkeley.

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math55-fall05-mt1-Bergman-soln - George M. Bergman Fall...

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