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About 11892 results. 1294 free access solutions
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| m97006 | The following is analogous to the "big-Oh" notation introduced in conjunction with Definition 5.23.
For f, g: Z+ → R we say that / is of order at least g if there exist constants M ∈ R+ and f: ∈ Z+ such that |f(n)| > Mg(n) for all n ∈ Z+, where n > k. In this case we write f ∈ Ω (g) and say that f is "big Omega of g." So Ω (g) represents the set of all functions with domain Z+ and codomain R that dominate g.
Suppose that f, g, h: Z+ → R, where f(n) = 5n2 + 3n, g(n) = n2, h(n) = n, for all n ∈ Z+. Prove that (a) f ∈ Ω (g); (b) g ∈ Ω (f); (c) f ∈ Ω(h); and (d) h ∈ Q(f) - that is, h is not "big Omega of f." |
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| m97009 | The following provides an alternative way to establish Lagrange s Theorem. Let G be a group of order n, and let H be a subgroup of G of order m.
(a) Define the relation R on G as follows: If a, b ∈ G, then a R b if a-1b ∈ H. Prove that R is an equivalence relation on G.
(b) For a, b ∈ G, prove that a R b if and only if aH = bH.
(c) If a ∈ G, prove that [a], the equivalence class of a under R, satisfies [a] = aH.
(d) For each a ∈ G, prove that |aH| = |H|.
(e) Now establish the conclusion of Lagrange s Theorem, namely that |H| divides |G|. |
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| m97010 | The following provides an outline for a proof of Theorem 17.18.
(a) Consider a parallel class of lines given by y = mx + b, where m ∈ F, m ≠ 0. Show that each line in this class intersects each "vertical" line and each "horizontal" line in exactly one point of AP (F). Thus the configuration obtained by labeling the points of AP (F), as in Figs. 17.4, 17.5, and 17.6, is a Latin square.
(b) To show that the Latin squares corresponding to two different classes, other than the classes of slope 0 or infinite slope, are orthogonal, assume that an ordered pair (i, j) appears more than once when one square is superimposed upon the other. How does this lead to a contradiction? |
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| m97011 | The following provides an outline for proving Corollary 8.2. Fill in the needed details.
(a) What is Et-1, and how are Lt and Lt-1 related?
(b) Show that
(c) For all 1 ≤ m ≤ t - 1, how are Lm, Lm+1, and Em related?
(d) Using the results in steps (a) through (d), establish the corollary by a backward type of induction. |
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| m97018 | The four input lines for the gating network shown in Fig. 15.12 provide the binary equivalents of the numbers 0, 1, 2, ..., 15, where each number is represented as abce, with e the least significant bit.
(a) Determine the d.n.f. of f, whose value is 1 for abce prime, and 0 otherwise.
(b) Draw the two-level gating network for f as a minimal sum of products.
(c) We are informed that the given network is part of a larger network and that, as a result, the binary equivalents of the numbers 10 through 15 are never provided as input. Design a two-level gating network for f under these circumstances. |
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| m97021 | The freshman class of a private engineering college has 300 students. It is known that 180 can program in Java, 120 in Visual BASIC+, 30 in C++, 12 in Java and C++, 18 in Visual BASIC and C++, 12 in Java and Visual BASIC, and 6 in all three languages.
(a) A student is selected at random. What is the probability that she can program in exactly two languages?
(b) Two students are selected at random. What is the probability that they can (i) both program in Java? (ii) both program only in Java? |
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| m97034 | The integer sequence a1, a2, a3, . . ., defined explicitly by the formula an = 5n for n e Z+, can also be defined recursively by
1) a1 = 5; and
2) an+1, an + 5, for n > 1.
For the integer sequence b1, b2, b3, . . . , where bn = n(n + 2) for all n ∈ Z+, we can also provide the recursive definition:
1) b1 = 3; and
2) bn+1 = bn + 2n + 3, for n > 1.
Give a recursive definition for each of the following integer sequences c1, c2, c3, . . ., where for all n ∈ Z+ we have
a) cn = 7n
b) cn = 7n
c) cn = 3n + 7
d) cn = 7
e) cn = n2
f) cn = 2 - (-1)n |
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| m97035 | The integer variables m and n are assigned the values 3 and 8, respectively, during the execution of a program (written in pseudocode). Each of the following successive statements is then encountered during program execution. [Here the values of m, n following the execution of the statement in part (a) become the values of m, n for the statement in part (b), and so on, through the statement in part (e).] What are the values of m, n after each of these statements is encountered?
(a) if n - m = 5 then n : = n - 2
(b) if ((2 * m = n) and (⌊n/4⌋ = 1)) then
n : = 4 * m - 3
(c) if ((n < 8) or (⌊m/2⌋ = 2)) then n : = 2 * m
else m : = 2 * n
(d) if ((m < 20) and (⌊n/6⌋ = 1)) then
m : = m - n - 5
(e) if ((n = 2 * m) or (⌊n/2⌋ = 5)) then
m : = m + 2 |
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| m97058 | The matrix transformation f: R2 → R2 defined by f(v) = Av, where
and A: is a real number, is called dilation if k > 1 and contraction if 0 < k < 1. Thus, dilation stretches a vector, whereas contraction shrinks it. If R is the rectangle defined in Exercise 2, find and sketch the image of R for
(a) k = 4;
(b) k = 1/4. |
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| m97059 | The matrix transformation f: R2 → R2 defined by f(v) = Av, where
and A: is a real number, is called dilation in the y-direction if k > 1 and contraction in the y-direction if 0 < k < 1. If R is the unit square and f is the contraction in the y-direction with k = 1/2, find and sketch the image of R. |
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| m97063 | The men s final at Wimbledon is won by the first player to win three sets of the five-set match. Let C and M denote the players. Draw a tree diagram to show all the ways in which the match can be decided. |
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| m97079 | The nine members of a coed intramural volleyball team are to be randomly selected from nine college men and ten college women. To be classified as coed the team must include at least one player of each gender. What is the probability the selected team includes more women than men? |
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| m97101 | The parking lot for a local restaurant has 41 parking spaces, numbered consecutively from 0 to 40. Upon driving into this lot, a patron is assigned a parking space by the parking attendant who uses the hashing function h(k) = & mod 41, where k is the integer obtained from the last three digits on the patron s license plate. Further, to avoid a collision (where an occupied space might be assigned), when such a situation arises, the patron is directed to park in the next (consecutive) available space - where 0 is assumed to follow 40.
(a) Suppose that eight automobiles arrive as the restaurant opens. If the last three digits in the license plates for these eight patrons (in their order of arrival) are
206, 807, 137, 444, 617, 330, 465, 905,
respectively, which spaces are assigned to the drivers of these eight automobiles by the parking attendant?
b) Following the arrival of the eight patrons in part (a), and before any of the eight could leave, a ninth patron arrives with a license plate where the last three digits are 00x. If this patron is assigned to space 5, what is (are) the possible value(s) of x? |
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| m97123 | The probability distribution for a random variable X is given by Pr(X = x) = (3x + l)/22, x = 0, 1, 2, 3. Determine
(a) Pr(X = 3).
(b) Pr{X < 1).
(c) Pr(l < X < 3).
(d) Pr(X > -2).
(e) Pr(X = 1X < 2). |
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| m97186 | The route that Jackie follows to school each day includes eight stoplights. When she reaches each stoplight, the probability that the stoplight is red is 0.25 and it is assumed that the stoplights are spaced far enough apart so as to operate independently. If the random variable Z counts the number of red stoplights Jackie encounters one particular day on her ride to school, determine
(a) Pr(X = 0).
(b) Pr(X = 3).
(c) Pr(X > 6).
(d) Pr(X > 6|Z > 4).
(e) E(X).
(f) Var(X). |
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| m97191 | The set W of all 2 ( 2 matrices A with trace equal to zero is a subspace of M22. Let
Show that span S = W. |
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| m97192 | The set W of all 2 ( 2 symmetric matrices is a subspace of M22. Find three 2 ( 2 matrices v1, v2, and v3 so that every vector in W can be expressed as a linear combination of v1, v2, and v3. |
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| m97193 | The set W of all 2 ( 3 matrices of the form
Where c = a + b, is a subspace of M23. Show that every vector in W is a linear combination of
And |
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| m97194 | The set W of all 3 ( 3 matrices A with trace equal to zero is a subspace of M33. Determine a subset S of W so that span S = W. |
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| m97219 | The three 4 × 4 Latin squares in Tables 17.3, 17.4, and 17.6 are orthogonal in pairs. Can you find another 4 × 4 Latin square that is orthogonal to each of these three? |
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