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Discrete and Combinatorial Mathematics An Applied Introduction 5th edition Ralph P. Grimaldi - Solutions

(a) Prove Theorem 16.4. (b) Extending the idea developed in Theorem 16.4 and Example 16.9 to the group Z6 × Z6 × Z6 = Z36 answer the following. (i) What is the order of this group? (ii) Find a subgroup of Z36 of order 6, one of order 12, and one of order 36. (iii) Determine the inverse of each of
(a) If H, K are subgroups of a group G, prove that H ∩ K is also a subgroup of G. (b) Give an example of a group G with subgroups H, K such that H ∪ K is not a subgroup of G.
(a) Find all x in (Z*5 ∙) such that x = x-1. (b) Find all x in (Z*11, ∙) such that x = x-1. (c) Let p be a prime. Find all x in (Z*p, ∙) such that x = x-1. (d) Prove that (p - 1)! ≡ - 1 (mod p), for p a prime. [This result is known as Wilson's Theorem, although it was only conjectured by
Prove parts (c) and (d) of Theorem 16.1. (c) If a, b, c ∈ G and ab = ac, then b = c. (Left-cancellation property) (d) If a, b, c ∈ G and ba = ca, then b = c. (Right-cancellation property)
(a) Find x in (Ug, ∙) where x ≠ 1, x ≠ 7 but x = x-1. (b) Find x in (U16, ∙) where x ≠ 1, x ≠ 15 but x = x-1. (c) Let k ∈ Z+, k ≥ 3. Find x in (U2k, ∙) where x ≠ 1, x ≠ 2k - 1 but x = x-1.
Let G = {q ∈ Q|q ≠ - 1}. Define the binary operation o on G by x ◦ y = x + y + xy. Prove that (G, ◦) is an abelian group.
Define the binary operation ◦ on Z by x ◦ y = x + y + 1. Verify that (Z, ◦) is an abelian group.
Let S = R* × R. Define the binary operation ◦ on S by (u, v) ◦ (x, y) = (ux, vx + y). Prove that (S, ◦) is a non-abelian group.
For any group G prove that G is abelian if and only if (ab)2 = a2b2 for all a, b ∈ G.
If G is a group, prove that for all a, b ∈ G, (a) (a-1)-1 = a (b) (ab)-1 = b-1a-1
Consider the configurations shown in Fig. 16.5.(a) Determine Ï€*2, Ï€*3, r*2, and r*4.(b) Verify that (Ï€1-1)* = (Ï€*1)-1 and (r3-1)* = (r3*)-1.(c) Verify that (Ï€1r1)* = Ï€*1r1* and (Ï€3r4)* = Ï€3*r4*.
Answer Exercise 11 for a 4 × 4 chessboard. [Replace each "nine" in part (b) with "sixteen."] (a) In how many ways can we paint the cells of a 3 × 3 chessboard using red and blue paint? (The back of the chessboard is black.) (b) In how many ways can we construct a 3 × 3 chessboard by joining
(a) Let S be a set of configurations and G a group of permutations that acts on S. If x ˆˆ S, prove that {Ï€ ˆˆ G|Ï€*(x) = x] is a subgroup of G (called the stabilizer of x).(b) Determine the respective stabilizer subgroups in part (a) for each of the
(a) Determine the order of each of the elements in Exercise 2. (b) State a general result about the order of an element in Sn in terms of the lengths of the cycles in its decomposition as a product of disjoint cycles.
(a) Determine the number of distinct ways one can color the vertices of an equilateral triangle using the colors red and white, if the triangle is free to move in three dimensions. (b) Answer part (a) if the color blue is also available.
Answer the questions in Exercise 4 for a regular pentagon. (a) Determine the number of distinct ways one can color the vertices of an equilateral triangle using the colors red and white, if the triangle is free to move in three dimensions. (b) Answer part (a) if the color blue is also available.
(a) How many distinct ways are there to paint the edges of a square with three different colors? (b) Answer part (a) for the edges of a regular pentagon.
We make a child's bracelet by symmetrically placing four beads about a circular wire. The colors of the beads are red, white, blue, and green, and there are at least four beads of each color, (a) How many distinct bracelets can we make in this way, if the bracelets can be rotated but not
A baton is painted with three cylindrical bands of color (not necessarily distinct), with each band of the same length. (a) How many distinct paintings can be made if there are three colors of paint available? How many for four colors? (b) Answer part (a) for batons with four cylindrical bands. (c)
In how many ways can we 2-color the vertices of the configurations shown in Fig. 16.9 if they are free to move in(a) Two dimensions?(b) Three dimensions?
Find the number of nonequivalent 4-colorings of the vertices in the configurations shown in Fig. 16.11 when they are free to move in(a) Two dimensions;(b) Three dimensions.
(a) In how many ways can we 3-color the vertices of a regular hexagon that is free to move in space? (b) Give a combinatorial argument to show that for all m ∈ Z+, (m6 + 2m + 2m2 + 4m3 + 3m4) is divisible by 12.
(a) In how many ways can we 5-color the vertices of a regular hexagon that is free to move in two dimensions? (b) Answer part (a) if the hexagon is free to move in three dimensions. (c) Find two 5-colorings that are equivalent for case (b) but distinct for case (a).
In how many distinct ways can we 3-color the edges in the configurations shown in Fig. 16.11 if they are free to move in(a) Two dimensions;(b) Three dimensions?
(a) In how many distinct ways can we 3-color the edges of a square that is free to move in three dimensions? (b) In how many distinct ways can we 3-color both the vertices and the edges of such a square? (c) For a square that can move in three dimensions, let k, m, and n denote the number of
(a) Find the pattern inventory for the 2-colorings of the edges of a square that is free to move in (i) Two dimensions; (ii) Three dimensions. (Let the colors be red and white.) (b) Answer part (a) for 3-colorings, where the colors are red, white, and blue.
If a regular pentagon is free to move in space and we can color its vertices with red, white, and blue paint, how many nonequivalent configurations have exactly three red vertices? How many have two red, one white, and two blue vertices?
Suppose that in Example 16.35 we 2-color the faces of the cube, which is free to move in space.(a) How many distinct 2-colorings are there for this situation?(b) If the available colors are red and white, determine the pattern inventory.(c) How many nonequivalent colorings have three red and three
For the organic compounds in Example 16.36, how many have at least one bromine atom? How many have exactly three hydrogen atoms?
Find the pattern inventories for the 2-colorings of the vertices in the configurations in Fig. 16.11, when they are free to move in space. (Let the colors be green and gold.)
(a) In how many ways can the seven (identical) horses on a carousel be painted with black, brown, and white paint in such a way that there are three black, two brown, and two white horses? (b) In how many ways would there be equal numbers of black and brown horses? (c) Give a combinatorial argument
(a) In how many ways can we paint the eight squares of a 2 × 4 chessboard, using the colors red and white? (The back of the chessboard is black cardboard.) (b) Find the pattern inventory for the colorings in part (a). (c) How many of the colorings in part (a) have four red and four white squares?
(a) In how many ways can we 2-color the eight regions of the pinwheel shown in Fig. 16.16, using the colors black and gold, if the back of each region remains grey?(b) Answer part (a) for the possible 3-colorings, using black, gold, and blue paints to color the regions.(c) For the colorings in part
Prove parts (b) and (c) of Theorem 16.5. f(an) = [f(a)n for all a ∈ G and all n ∈ Z.
(a) Determine U14, the group of units for the ring (Z14, +, ∙). (b) Show that U14 is cyclic and find all of its generators.
For a group G, prove that the function f: G → G defined by f(a) = a-1 is an isomorphism if and only if G is abelian.
If f: G → H, g: H → K are homomorphisms, prove that the composite function g ◦ f: G → K, where (g ◦ f)(x) = g(f(x)), is a homomorphism.
For to = (1/√2)(1 + i), let G be the multiplicative group {wn|n ∈ Z+, 1 ≤ n ≤ 8}. (a) Show that G is cyclic and find each element x e G such that (x) = G. (b) Prove that G is isomorphic to the group (Z8, +).
(a) Find all generators of the cyclic groups (Z12, +), (Z16, +), and (Z24, +). (b) Let G = (a) with σ(a) = n. Prove that ak, k ∈ Z+, generates G if and only if k and n are relatively prime. (c) If G is a cyclic group of order n, how many distinct generators does it have?
Let f: G → H be a group homomorphism. If a ∈ G with σ(a) = n, and σ(f(a)) = k (in H), prove that k|n.
(a) Determine A2, A3, and A4.(b) Verify that {A, A2, A3, A4} is an abelian group under ordinary matrix multiplication.(c) Prove that the group in part (b) is isomorphic to the group shown in Table 16.6.Let
If G = (Z6, +), H = (Z3, +), and K = (Z2, +), find an isomorphism for the groups H × K and G.
Let f: G → H be a group homomorphism onto H. If G is abelian, prove that H is abelian.
Let (Z ≠ Z, ⊕) be the abelian group where (a, b) ⊕ (c, d) = (a + c, b + d) - here a + c and b + d are computed using ordinary addition in Z - and let (G, +) be an additive group. If f (1, 3) = g1 group homomorphism where f(1, 3) = g1 and f(3, 7) = g2, express f(4, 6) in terms of g1 and g2.
Let f: (Z × Z, ⊕) → (Z, +) be the function defined by f(x, y) = x - y. [Here (Z × Z, ⊕) is the same group as in Exercise 5, and (Z, +) is the group of integers under ordinary addition.] (a) Prove that f is a homomorphism onto Z. (b) Determine all (a, b) ∈ Z × Z with f(a, b) = 0. (c) Find
Find the order of each element in the group of rigid motions of (a) The equilateral triangle; and (b) The square.
In S5 find an element of order n, for all 2 ≤ n ≤ 5. Also determine the (cyclic) subgroup of S5 that each of these elements generates.
Let G = S4. (a) ForFind the subgroup H = (Î±), (Î²) Determine the left cosets of H in G.
Prove Corollaries 16.1 and 16.2. Corollaries 16.1 If G is a finite group and a ∈ G, then 0(a) divides |G|. Corollaries 16.2 Every group of prime order is cyclic.
Let H and K be subgroups of a group G, where e is the identity of G. Prove that if |H| = 10 and |K| = 21, then H ∩ K = {e}.
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,
(a) Fermat's Theorem. If p is a prime, prove that ap ≡ a (mod p) for each a ∈ Z. [How is this related to Exercise 22(a) of Section 14.3?](b) Euler's Theorem. For each n ∈ Z+, n > 1, and each a ∈ Z, prove that if gcd(a, n) = 1, then a
Answer Exercise 1 for the case where Î± is replaced by
For G = (Z24, +), find the cosets determined by the subgroup H = ([3]). Do likewise for the subgroup K = ([4]).
Let G be a group with subgroups H and K. If |G| = 660, |K| = 66, and K ⊆ H ⊂ G, what are the possible values for |H|?
Let R be a ring with unity u. Prove that the units of R form a group under the multiplication of the ring.
Let G = S4, the symmetric group on four symbols, and let H be the subset of G where(a) Construct a table to show that H is an abelian subgroup of G. (b) How many left cosets of H are there in G? (c) Consider the group (Z2 Ã— Z2, Š•) where (a, b) Š• (c, d) = (a + c, b
Let p be a prime, (a) If G has order 2p, prove that every proper subgroup of G is cyclic, (b) If G has order p2, prove that G has a subgroup of order p.
Determine the cipher-text for the plaintext INVEST IN STOCKS, when using RSA encryption with e = 1 and n = 2573.
Determine the cipher-text for the plaintext ORDER A PIZZA, when using RSA encryption with e = 5 and n = 1459.
Determine the plaintext for the RSA cipher-text 1418 1436 2370 1102 1805 0250, if e = 11 and n = 2501.
Determine the plaintext for the RSA cipher-text 0986 3029 1134 1105 1232 2281 2967 0272 1818 2398 1153, if e = 17 and n = 3053.
Find the primes p, q if n = pq = 121,361 and
Find the primes p, q if n = pq = 5,446,367 and
A binary symmetric channel has probability p = 0.05 of incorrect transmission. If the code word c = 011011101 is transmitted, what is the probability that (a) We receive r = 011111101? (b) We receive r = 111011100? (c) A single error occurs? (d) A double error occurs? (e) A triple error occurs? (f)
Let E: Z32 → Z92 be the encoding function for the (9, 3) triple repetition code. (a) If D: Z92 → Z32 is the corresponding decoding function, apply D to decode the received words (i) 111101100; (ii) 000100011; (iii) 010011111. (b) Find three different received words r for which D(r) = 000. (c)
The (5m, m) five-times repetition code has encoding function E: Zm2 → Z5m2, where E(w) = wwwww. Decoding with D: Z5m2 → Zm2 is accomplished by the majority rule. (Here we are able to correct single and double errors made in transmission.) (a) With p = 0.05, what is the probability for the
For Example 16.24, list the elements in 5(101010, 1) and 5(111111, 1).
(a) Show that the 1 × 9 matrix G = [1 1 1 ... 1] is the generator matrix for the (9, 1) nine times repetition code. (b) What is the associated parity-check matrix H in this case?
Given n ˆˆ Z+, let the set M(n, k) Š† Zn2 contain the maximum number of code words of length n, where the minimum distance between code words is 2k + 1. Prove that(The upper bound on |M(n, k)| is called the Hamming bound; the lower bound is referred to as the Gilbert bound.)
Decode each of the following received words for Example 16.24. (a) 110101 (b) 101011 (c) 001111 (d) 110000
(a) If x ∈ Z102, determine |S(x, 1)|, |S(x, 2)|, |S(x, 3)|. (b) For n, k ∈ Z+ with 1 ≤ k ≤ n, if x ∈ Zn2, what is |S(x, k)|?
For each of the following encoding functions, find the minimum distance between the code words. Discuss the error-detecting and error-correcting capabilities of each code. (a) E: Z22 → Z52 00 → 00001 01 → 01010 10 → 10100 11 → 11111 (b) E: Z22 → Z102 00 → 0000000000 01 →
(a) Use the parity-check matrix H of Example 16.25 to decode the following received words. (i) 111101 (ii) 110101 (iii) 001111 (iv) 100100 (v) 110001 (vi) 111111 (vii) 111100 (viii) 010100 (b) Are all the results in part (a) uniquely determined?
The encoding function E: Z22 †’ Z52 is given by the generator matrix(a) Determine all code words. What can we say about the error-detection capability of this code? What about its error- correction capability? (b) Find the associated parity-check matrix H. (c) Use H to decode each of the
Define the encoding function E: Z32 †’ Z62 by means of the parity-check matrix(a) Determine all code words. (b) Does this code correct all single errors in transmission?
Let E: Z82 → Z122 be the encoding function for a code C. How many calculations are needed to find the minimum distance between code words? How many calculations are needed if E is a group homomorphism?
(a) Use Table 16.9 to decode the following received words.000011 100011 111110 100001001100 011110 001111 111100(b) Do any of the results in part (a) change if a different set of coset leaders is used?
(a) Construct a decoding table (with syndromes) for the group code given by the generator matrix(b) Use the table from part (a) to decode the following received words. 11110 11101 11011 10100 10011 10101 11111 01100
(a) Encode the following messages:1000 1100 1011 1110 1001 1111.(b) Decode the following received words:1100001 1110111 0010001 0011100.(c) Construct a decoding table consisting of the syndromes and coset leaders for this code.(d) Use the result in part (c) to decode the received words given in
(a) Let p = 0.01 be the probability of incorrect transmission for a binary symmetric channel. If the message 1011 is sent via the Hamming (7, 4) code, what is the probability of correct decoding? (b) Answer part (a) for a 20-bit message sent in five blocks of length 4.
Let f: G → H be a group homomorphism with eH the identity in H. Prove that (a) K = {x ∈ G|f(x) = eH] is a subgroup of G. (K is called the kernel of the homomorphism.) (b) if g ∈ G and x ∈ K, then gxg-1 ∈ K.
If G, H, and K are groups and G = H × K, prove that G contains subgroups that are isomorphic to H and K.
Let G be a group where a2 = e for all a ∈ G. Prove that G is abelian.
If G is a group of even order, prove that there is an element a ∈ G with a ≠ e and a = a-1.
Let f: G → H be a group homomorphism onto H. If G is a cyclic group, prove that H is also cyclic.
(a) Consider the group (Z2 × Z2, ⊕) where, for a, b, c, d ∈ Z2, (a, b) ⊕ (c, d) = (a + c, b + d) - the sums a + c and b + d are computed using addition modulo 2. What is the value of (1, 0) ⊕ (0, 1) ⊕ (1, 1) in this group? (b) Now consider the group (Z2 × Z2 × Z2, ⊕) where (a, b, c)
Let (G, ◦) be a group where x ◦ a ◦ y = b ◦ a ◦ c = x ◦ y = b ◦ c, for all a, b, c, x, y ∈ G. Prove that (G, ◦) is an abelian group.
Let f(x), g(x) ∈ Z7[x] where f(x) = 2x4 + 2x3 + 3x2 + x + 4 and g(x) = 3x3 + 5x2 + 6x + 1. Determine f(x) + g(x), f(x) - g(x), and f(x)g(x).
For each of the following polynomials f(x)∈Z7[x], determine all of the roots in Z7 and then write fix) as a product of first-degree polynomials. (a) f(x) = x3 + 5x2 + 2x + 6 (b) f(x) = x7 - x
Given a field F, let f(x) ∈ F[x] where f(x) = anxn + an-1xn-l + • • • + a2x2 + a1x + do. Prove that x - 1 is a factor of f(x) if and only ifan + an - l + ∙∙∙∙∙∙∙ + a2 + a1 + a0 = 0.
Let R, S be rings, and let g: R †’ S be a ring homomorphism. Prove that the function G: R[x] †’ S[x] defined byis a ring homomorphism
If R is an integral domain, prove that if fix) is a unit in R[x], then fix) is a constant and is a unit in R.
Verify that f(x) = 2x + 1 is a unit in Z4[x]. Does this contradict the result of Exercise 14?
For n ∈ Z+, n > 2, let f(x) ∈ Zn[x]. Prove that if a, b ∈ Z and a = b (mod n), then f(a) = f(b) (mod n).
If F is a field, let S ⊂ F[x] where f(x) = anxn + an-1xn-l + ∙∙∙∙∙∙ + a2x2 + a1x + a0 ∈ S if and only if an + an-1 + ∙∙∙∙∙∙∙ + a2 + a1 + a0 = 0. Prove that S is an ideal of F[x].
Let (R, +, •) be a ring. If I is an ideal of R, prove that I[x], the set of all polynomials in the indeterminate x with coefficients in f, is an ideal in R[x].
Complete the proofs of Theorem 17.1 and Corollary 17.1.Let R[x] be a polynomial ring.a) If R is commutative, then R [x] is commutative.b) If R is a ring with unity, then R[x] is a ring with unity.c) R[x] is an integral domain if and only if R is an integral domain.If R is a ring, then under the
For each of the following pairs f(x), g(x), find q(x), r(x) so that g(x) = q(x)f(x) + r(x), where r(x) = 0 or degree r(x) < degree f(x). a) f(x), g(x) ∈ Q(x), f(x) x4 - 5x3 + 7x, g(x) = x5 - 2x2 + 5x - 3 b) f(x), g(x) ∈ Z2[x], f(x) = x2 + 1, g(x) = x4 + x3 + x2 + x + l c) f(x), g(x) ∈ Z5[x],
(a) If f(x) = x4 - 16, find its roots and factorization in Q[x]. (b) Answer part (a) for f(x) ∈ R[a], (c) Answer part (a) for f(x) ∈ C[x], (d) Answer parts (a), (b), and (c) for f(x) = x4 - 25.

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