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Questions and Answers of
Linear Algebra
For n ∈ Z+, n > 2, prove that for any statements p1, p2, ∙ ∙ ∙ pn (a) ¬(p1 ∨ p2 ∨ ∙ ∙ ∙ ∨ pn) ⇔ ¬Pl ∧ -P2 ∧ ∙ ∙ ∙ ∧ ¬pn. (b) ¬ (p1 ∧ p2 ∧ ∙ ∙ ∙ ∧ pn)
(a) Give a recursive definition for the intersection of the sets A1, A2, . . . , An, An+1 ⊂ °U, n > 1. (b) Use the result in part (a) to show that for all n, r ∈ Z+ with n > 3 and 1 < r < n, (A1
For n > 2 and any sets A1, A2, . . . , An °U, prove that
Use the result of Example 4.17 to show that if sets A, B1, B2, . . ., Bn ⊂ °U and n > 2, then A ∩ (B1 ∪ B2 ∪ ∙ ∙ ∙ ∪ Bn) = (A ∩ B1) ∪ (A ∩ B2) ∪ ∙ ∙ ∙ ∪ (A ∩ Bn).
(a) Develop a recursive definition for the addition of n real numbers x1, x2, .. ., xn, where n > 2. (b) For all real numbers x1, x2, and x3, the associative law of addition states that x1 + (x2 +
(a) Develop a recursive definition for the multiplication of n real numbers x1, x2, . . . , xn, where n > 2. (b) For all real numbers x1, x2, and x3, the associative law of multiplication states that
If A = {1, 2, 3, 4}, B = {2, 5}, and C = {3, 4, 7}, determine A × B; (B × A); A U (B × C); (A × B) × C; (A × C) U (B × C).
For A, B, C ⊂s °U, prove that A × (B - C) = (A × B) - (A × C).
Let
(a) Give a recursive definition for the relation R ⊂ Z+ × Z+ where (m, n) ∈ R if (and only if) m > n. (b) From the definition in part (a) verify that (5, 2) and (4, 4) are in 31.
If A = {1, 2, 3}, and B = {2, 4, 5}, give examples of (a) three nonempty relations from A to B; (b) three nonempty relations on A.
For A, B as in Exercise 2, determine the following: (a) | A X B |; (b) the number of relations from A to B; (c) the number of relations on A; (d) the number of relations from A to B that contain
Let A, B, C, Z) be nonempty sets.(a) Prove that A × B ⊂ C×D) if and only if A ⊂ C and B⊂D.(b) What happens to the result in part (a) if any of the sets A, B, C, D is empty?
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
(a) If A = {1, 2, 3, 4, 5} and B = {w, x, y, z}, how many elements are there in P(A X B)?(b) Generalize the result in part (a).
Logic chips are taken from a container, tested individually, and labeled defective or good. The testing process is continued until either two defective chips are found or five chips are tested in
Complete the proof of Theorem 5.1.
Determine whether or not each of the following relations is a function. If a relation is a function, find its range. (a) {(x, y)\x, y ∈ Z, y = x2 + 7}, a relation from Z to Z (b) {(x, y)|x, y ∈
Determine all x ∈ R such that [x] + [x + 1/2] + = [2x].
For n, k ∈ Z+, prove that [n/k] = [n - 1)/k] + 1.
(a) Let a ∈ R+ where a > 1. Prove that (i) [a] /a = 1; and (ii) [|a] /a! = 1(b) If a ∈ R+ and 0 < a < 1, which result(s) in part (a) is are) true?
Let a1, a2, a3,..... be the integer sequence defined recursively by(1) a1 = 1; and(2) For all n ∈ Z+ where n > 2, an = 2a [n/2](a) Determine an for all 2 < n < 8.(b) Prove that an < n
For each of the following functions, determine whether it is one-to-one and determine its range. (a) f: Z→Z, f (x) = 2x + 1 (b) f: Q→Q.f(x) = 2x + 1 (c) f: Z → Z, f(x) = x3 - x (d) f: R→R.
Let /: R R where f(x) = x2. Determine /(A) for the following subsets A taken from the domain R. (a) A = {2, 3} (b) A = {-3, -2, 2, 3} (c) A = (-3, 3) (d) A = (-3, 2] (e) A = [-7, 2] (f) A = (-4,
Let A = {1, 2, 3, 4, 5}, B = {w, x, y, z], A1 = {2, 3, 5} ⊂ A, and g: A1 → B. In how many ways can g be extended to a function f: A → B?
Give an example of a function f: A → B and A1, A2 ⊂ A for which f(A1 ∩ A2) ≠ f(A1) ∩ (A2). [Thus the inclusion in Theorem 5.2(b) may be proper.]
Prove parts (a) and (c) of Theorem 5.2.
Does the formula f(x) - 1/{x2 - 2) define a function f: R → R? function f: Z → R?
Let f : A → B, where A = X ∪ Y with X ∩ Y = 0. If f|x and f|Y are one-to-one, does it follow that f is one-to-one?
For n ∈ Z+ define Xn = {1, 2, 3, . . . , n}. Given m, n ∈ Z+, f: Xm → Xn is called monotone increasing if for all i, j ∈ Xm, 1 < i < j < m => f(i) < f(j). (a) How many monotone
Determine the access function f(aij), as described in Example 5.10(d), for a matrix A = {alj)m × n, where (a) m = 12, n = 12; (b) m = 7, n = 10; (c) m = 10, n = 7.
(a) Let A be an m Xn matrix that is to be stored (in a contiguous manner) in a one-dimensional array of r entries. Find a formula for the access function if an is to be stored in location k ( > 1) of
The following exercise provides a combinatorial proof for a summation formula we have seen in four earlier results: (1) Exercise 22 in Section 1.4; (2) Example 4.4; (3) Exercise 3 in Section 4.1; and
One version of Ackermann's function A(m,n) is defined recursively for m, n ∈ N byA(0, n) = n + 1, n > 0;A(m, 0) = A(m - 1, 1), m > 0; and A(m, n) = A(m - 1, A(m, n - 1)), m, n > 0.[Such
Given sets A, B, we define a partial function f with domain A and codomain B as a function from Af to B, where ϕ ≠ A'⊂ A. [Here f(x) is not defined for x ∈ A - A'.] For example, f : R* → R,
Let A = {1, 2, 3, 4} and B = {x, y, z}. (a) List five functions from A to B. (b) How many functions f: A → B are there? (c) How many functions f:A→B are one-to-one? (d) How many functions g: B
Let A, B, C c R2 where A = {(*, y)|y = 2x + 1}, B = {(x, y)|y = 3x}, and C = {Qt, y)|jc - y = 7}. Determine each of the following:(a) A ∩ B (b) B ∩ C(c) A- ∪ C- (d) B- ∪ C-
Let A, B, C ⊂ Z2 where A = {(x, y)|y = 2x + 1}, B = {(x, y)|y = 3x}, and C = {(x, y)|x - y = 7}. (a) Determine (i) A ∩ B iii) B ∩ C iii) A ∪ C (iv) B ∪ C (b) How are the answers for
Determine each of the following: (a) [2.3 - 1.6] (b) |2.3| - |1.6| (c) [3.4][6.2] d) [3.4][6.21] (e) [2π] (f) 2[π]
Determine whether each of the following statements is true or false. If the statement is false, provide a counterexample. (a) [a] = [a] for all a ∈ Z. (b) [a] = [a] for all ∈ R. (c) [a] = [a] - 1
Find all real numbers x such that (a) 7[x] = [7x] (b) [7x] = 7 (c) [x + 7 ] = x + 7 d) [x + 7] = [x] + 7
Give an example of finite sets A and B with | A |, | B | > 4 and a function f : A → B such that (a) f is neither one-to-one nor onto; (b) f is one-to-one but not onto; (c) f is onto but not one-to
Suppose we have seven different colored balls and four containers numbered I, II, III, and IV. (a) In how many ways can we distribute the balls so that no container is left empty? (b) In this
Determine the next two rows (m = 9, 10) of Table 5.1 for the Stirling numbers S(m, n), where 1 < n < m.
a) In how many way scan 31,100,905 be factored into three factors, each greater than 1, if the order of the factors is not relevant?b) Answer part (a), assuming the order of the three factors is
a) How many two-factor unordered factorizations, where each factor is greater than 1, are there for 156,009?b) In how many ways can 156,009 be factored into two or more factors, each greater than 1,
Write a computer program (or develop an algorithm) to compute the Stirling numbers S{m,n) when 1 < m < 12 and 1 < n < m.
A lock has n buttons labeled 1, 2, . . . , n. To open this lock we press each of the n buttons exactly once. If no two or more buttons may be pressed simultaneously, then there are n\ ways to do
At St. Xavier High School ten candidates C1, C2, ..., C10, run for senior class president.(a) How many outcomes are possible where (i) there are no ties (that is, no two, or more, candidates receive
Form, n, r ˆˆ Z+ with m > m,rn, let Sr (m, n) denote the number of ways to distribute m distinct objects among n identical containers where each container receives at least r of the objects.
We use s(m,n) to denote the number of ways to seat m people at n circular tables with least one person at each table. The arrangements at any one table are not distinguished if one can be rotated
As in the previous exercise, s(m, n) denotes a Stirling number of the first kind.(a) For m > n > 1 prove thats(m,n) = (m - 1)s(m - 1,n) + (s(m - 1, n - 1).(b) Verify that for m > 2,
For each of the following functions /: Z -> Z, determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range /(Z).(a) f{x) = x + 7 (b) fix)
For each of the following functions g: R -► R, determine whether the function is one-to one and whether it is onto. If the function is not onto, determine the range g(R) a) gix) = x + 7 (b) gix) =
Verify that
a. Verify that 57b. Provide a combinatorial argument to prove that for all m, n ˆˆ Z+,
a. Let A = {1, 2, 3, 4, 5, 6, 7} and B = {v, w, x, y, z}. Determine the number of functions /: A → B where (i) f(A) = {v, x};(ii) |f(A)| = 2;(iii) f(A) = {w, x, y};(iv) If(A) | = 3; (v) f(A) = {u,
A chemist who has five assistants is engaged in a research project that calls for nine compounds that must be synthesized. In how many ways can the chemist assign these syntheses to the five
Use the fact that every polynomial equation having real- number coefficients and odd degree has a real root in order to show that the function f: R → R, defined by f{x) = x5 - 2x2 + x, is an onto
For A = {a, b, c], let f: A Ã A be the closed binary operation given in Table 5.6. Give an example to show that f is not associative.
State a result that generalizes the ideas presented in the previous two exercises.
For ϕ A⊂ Z+, let f, g: A × A → A be the closed binary operations defined by f(a, b) = min {a, b} and g(a, b) = max{a, b}. Does f have an identity element? Does g?
Let A = B = R. Determine πA(D) and πB(D) for each of the following sets D ⊂ A × B. (a) D = {{x, y)\x = y2} (b) D = {(x, y)|y = sin x} (c) D = {(x, y)|x2 + y2 = 1
Let Aj, 1 < i < 5, be the domains for a table D ⊂ A1 × A2 × A3 × A4 × A5, where A1 = {U, V, W, X, Y, Z} (used as code names for different cereals in a test), and A2 = A3 = A4 = A5 = Z+. The
Let A, 1 < i < 5, be the domains for a table D ⊂ A1 × A2 × A3 × A4 × A5, where A1 = {1,2} (used to identify the daily vitamin capsule produced by two pharmaceutical companies), A2 = {A, D, E},
Let /: R XR^Zbe the closed binary operation defined by f(a,b) = [a + b]. (a) Is f commutative? (b) Is f associative? (c) Does f have an identity element?
Each of the following functions f: Z × Z → Z is a closed binary operation on Z. Determine in each case whether f is commutative and/or associative.(a) f(x, y) = x + y - xy(b) f(x,y) = max{x, y},
Which of the closed binary operations in Exercise 3 have an identity?
Let |A| = 5. (a) What is |A × A|? (b) How many functions f: A × A are there? (c) How many closed binary operations are there on A? (d) How many of these closed binary operations are commutative?
Let A = {x, a, b, c, d}. (a) How many closed binary operations f on A satisfy f(a, b) = c? (b) How many of the functions f in part (a) have x as an identity? (c) How many of the functions f in part
Let A = {2, 4, 8, 16, 32}, and consider the closed binary operation f: A × A → A where f(a, b) = gcd (a, b). Does f have an identity element?
For distinct primes p, q let A = {pmqn}0 < m < 31, 0 < n < 37}. (a) What is |A|? (b) If f: A × A → A is the closed binary operation defined by f(a, b) = gcd (a, b), does f have an identity element?
Let triangle ABC be equilateral, with AB = 1. Show that if we select 10 points in the interior of this triangle, there must be at least two whose distance apart is less than 1/3.
Let ABCD be a square with AB = 1. Show that if we select five points in the interior of this square, there are at least two whose distance apart is less than 1 √2.
Let A ⊂ {1, 2, 3, ..., 25} where |A| = 9. For any subset B of A let sB denote the sum of the elements in B. Prove that there are distinct subsets C, D of A such that |C| = |D| =5 and sc = sD.
Let S be a set of five positive integers the maximum of which is at most 9. Prove that the sums of the elements in all the nonempty subsets of S cannot all be distinct.
During the first six weeks of his senior year in college, Brace sends out at least one resume each day but no more than 60 resumes in total. Show that there is a period of consecutive days during
Let S ⊂ Z+ with |S| =7. For ϕ A ⊂ S, let sA denote the sum of the elements in A. If m is the maximum element in S, find the possible values of m so that there will exist distinct subsets B, C
Let k ∈ Z+. Prove that there exists a positive integer n such that k\n and the only digits in n are O's and 3's.
(a) Find a sequence of four distinct real numbers with no decreasing or increasing subsequence of length 3.(b) Find a sequence of nine distinct real numbers with no decreasing or increasing
The 50 members of Nardine's aerobics class line up to get their equipment. Assuming that no two of these people have the same height, show that eight of them (as the line is equipped from first to
For k, n ∈ Z+, prove that if kn + 1 pigeons occupy n pigeonholes, then at least one pigeonhole has k + 1 or more pigeons roosting in it.
a) Let S ⊂ Z+. What is the smallest value for | S\ that guarantees the existence of two elements x, y ∈ S where x and y have the same remainder upon division by 1000?b) What is the smallest value
For m, n ∈ Z+, prove that if m pigeons occupy n pigeonholes, then at least one pigeonhole has [m - 1)/n[ + more pigeons roosting in it.
Let p1, p2, . .. , pn ∈ Z+. Prove that if p1 + P2 + ∙∙∙∙∙∙∙ + pn - n + 1 pigeons occupy n pigeonholes, then either the first pigeonhole has p1 or more pigeons roosting in it, or the
Let S = {3, 7, 11, 15, 19, ... , 95, 99, 103}. How many elements must we select from S to insure that there will be at least two whose sum is 110?
(a) Prove that if 151 integers are selected from {1, 2, 3, ..., 300}, then the selection must include two integers x, y where x|y or y|x.(b) Write a statement that generalizes the results of part (a)
(a) Show that if any 14 integers are selected from the set S = {1, 2, 3, ..., 25}, there are at least two whose sum is 26.(b) Write a statement that generalizes the results of part (a) and Example
(a) If S ⊂ Z+ and |S| >3, prove that there exist distinct x, y e S where x + y is even.(b) Let S ⊂ Z+ × Z+. Find the minimal value of |S| that guarantees the existence of distinct ordered
(a) If 11 integers are selected from {1, 2, 3, ..., 100}, prove that there are at least two, say x and y, such that 0 < |√x - √y < l.(b) Write a statement that generalizes the result of
(a) For A = {1, 2, 3, 4, . .., 7}, how many bijective functions f: A → A satisfy /(1) ≠ 1?(b) Answer part (a) where A = {x| x ∈ Z+, 1 < x < n}, for some fixed n ∈ Z+.
For each of the following functions f: R → R, determine whether f is invertible, and, if so, determine f-1. (a) f = {(x, y)|2x + 3y = 7} (b) f = {(x, y)\ax + by = c,b f 0} (c) f = {(x,y)\y =x3} (d)
Prove Theorem 5.9. If f: A -> B, g: B -> C are invertible functions, then g o f: A -> C is invertible and (g 0 f)-1 = f-1 °g-1.
If A = {1, 2, 3, 4, 5, 6, 7}, fl = {2, 4, 6, 8, 10, 12}, and f:A → B where f = {(1, 2), (2, 6), (3, 6), (4, 8), (5, 6), (6, 8), (7, 12)}, determine the preimage of B1 under f in each of the
Let f: R †’ R be defined by(a) Find f-1(-10), f-1(0), f-1(4), f-1(6), f-1(7), and f-1(8).(b) Determine the preimage under f for each of the intervals (i) [-5, -1}; (ii) [-5, 0}; (iii) [-2, 4};
Let f: R -> R be defined by f(x) = x2. For each of the following subsets B of R, find f-l(B). (a) B = {0, 1} (b) B = {-1, 0, 1} (c) B = [0, 1} (d) B = [0, 1) (e) B = [0, 4} (f) B = (0, 1} U (4, 9)
Let A = {1, 2, 3, 4, 5} and B = {6, 7, 8, 9, 10, 11, 12}. How many functions f:A^B are such that f-1({6, 7, 8}) = {1,2}?
Let f: R be defined by f(x)= [x], the greatest integer in x. Find f~l(B) for each of the following subsets B of R. (a) B = {0, 1} (b) B = {-1, 0, 1} (c) B = [0, 1) (d) B = [0, 2) (e) B = [-1, 2}
Let f, g: Z+ → Z+ where for all x e Z+, f(x) = x + 1 and g(x) = max{l, x - 1}, the maximum of 1 and x - f. (a) What is the range of /? (b) Is f an onto function? (c) Is the function f
Let f, g, h denote the following closed binary operations on P(Z+). For A, B ⊂ Z+, f(A, B) = A ∩ B, g(A, B) = A∪ B, h(A, B) = A△B.a) Are any of the functions one-to-one?b) Are any of f, g,
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