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Questions and Answers of
Linear Algebra
Prove that the sum of the cubes of three consecutive integers is divisible by 9.
Determine the last digit in 355.
For ra, n, r Z+, let p(m, n, r) count the number of partitions of ra into at most n (positive) summands each no larger than r. EvaluateP(k(n + 1), n, n), n Z+.
Given a ring (R, +, •)> an element r ∈ R is called idempo- tent when r2 = r. lf n ∈ Z+ with n > 2, prove that if k ∈ Zn and k is idempotent, then n - k + 1 is idempotent.
For the hashing function at the end of Example 14.17, find (a) h(123-04-2275); (b) a social security number n such that h(n) = 413, thus causing a collision with the number 081-37- 6495 of the
Write a computer program (or develop an algorithm) that implements the hashing function of Exercise 35.
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
Solve the following linear congruences for x.(a) 3x = 1 (mod 31) (b) 5x = 8 (mod 37)(c) 6x = 97 (mod 125)
Prove that if a, b, c, n ∈ Z with a, n > 0, and b = c (mod n), then ab = ac (mod an).
Let a, b, m, n ∈ Z with m, n > 0. Prove that if a = b (mod n) and m\n, then a = b (mod m).
Let m, n e Z+ with gcd(m, n) = 1 and let a, b ∈. Prove that a = b (mod m) and a = b (mod n) if and only if a = b (mod mn).
Provide a counterexample to show that the result in the preceding exercise is false if gcd(m, n) > 1
Prove that for all integers n exactly one of n, 2n - 1, and 2n + 1 is divisible by 3.
If n Z+ and n >2 prove that
(a) How many units are there in Z15? How many in Z3 x Z5? (b) Are Z15 and Z3 x Z5 isomorphic?
If f: R → S is a ring homomorphism and J is an ideal of 5, prove that f-l(J) = [a ∈ R|f(a) ∈ J} is an ideal of R.
Find a simultaneous solution for the system of two congruences: x = 5 (mod 8) x=73 (mod 81).
A band of 17 pirates captures a treasure chest full of (identical) gold coins. When the coins are divided up into equal numbers, three coins remain. One pirate accuses the distributor of miscounting
Find a simultaneous solution for the system of four congruences: x == l (mod 2) x = 2 (mod 3) x = 3 (mod 5) x = 5 (mod 7).
Complete the proofs of Theorems 14.15 and 14.16.
If R, 5, and T are rings and f:R → S, g: S → T are ring homomorphisms, prove that the composite function g o f: R → T is a ring homomorphism.
If 5 =then S is a ring under matrix addition and multiplication. Prove that R is isomorphic to S.
(a) Let(R, +, ∙)and(S, ⊕, ⊙) be rings with zero elements ZR and Zs, respectively. If f: R → S is a ring homomorphism, let K = [a ∈ R | f(a) = Zs}• Prove that K is an ideal of R. (K is
Use the information in Table 14.11 to compute each of the following in Z30.(a) (13)(23) + 18 (b) (11)(21) - 20(c) (13 + 19)(27) (d) (13)(29) + (24)(8)
(a) Construct a table (as in Example 14.21) for the isomorphism f: Z20 Z4 × Z5.(b) Use the table from part (a) to compute the following in Z20-(i) (17)(19) + (12)(14)(ii) (18)(11) - (9)(15)
Determine whether each of the following statements is true or false. For each false statement give a counterexample. (a) If (/?, +, •) is a ring, and 0 ≠ S ⊂ R with S closed under + and ∙,
(a) If p is a prime, prove that p divides (pk), for all 0 < k < p.(b) If a, b ∈ Z, prove that (a + b)p = ap + bp (mod p).
Given n positive integers x1, x2, ..........xn not necessarily distinct, prove that either n(x1 + x2 + ∙∙∙∙∙∙∙∙+ xt), for some 1 < i < n, or there exist 1 < i < j < n such that
Consider the ring (Z3, ⊕, ⊙) where addition and multiplication are defined by (a, b, c) ⊕ (d, e, f) = (a + d, b + e, c + f) and (a, b, c) ⊙ (d, e, f) = (ad, be, cf). (Here, forex- ample, a +
a) In how many ways can one select two positive integers m,n, not necessarily distinct, so that 1 < m < 100, 1 < n < 100 and the last digit of 7m + 3n is 8?b) Answer part (a) for the case
Let n ˆˆ Z+ with n > 1.a) If n = 2k where k is an odd integer, prove thatk3 =k (mod n).b) If n = 4k for some k ˆˆ Z+, prove that(2k)2 = 0 (mod ft).c) Prove that
Suppose that a, b, c ∈ Z and 5|(a2 + b2 + c2). Prove that 5|a or 5\b or 5|c.
Write a computer program (or develop an algorithm) that reverses the order of the digits in a given positive integer. For example, the input 1374 should result in the output 4731.
Suppose that a, b, k ∈ Z+ with a - b - p1e1p2e2 ∙∙∙∙∙∙∙∙ pkek, for P1, P2, ∙∙∙∙∙∙∙∙ Pk prime and e1, e2, . .. , ek ∈ Z+. For how many values of n ( > 1) is a = b
As the co-chairs of the Homecoming Parade Committee, Jerina and Noor must organize the freshmen for a pregame presentation. When they arrange these students in rows of 8, there are three students
Prove that a ring R is commutative if and only if (a + b)2 = a2 + 2ab + b2, for all a, b ∈ R.
A ring R is called Boolean if a2 = a for all a ∈ R. If R is Boolean, prove that (a) a + a = 2a = z, for all a ∈ R ', and (b) R is commutative.
With C the field of complex numbers and S the ring of 2 Ã 2 real matrices of the formdefine f: C S by f(a + bi) = for a + bi C. Prove that f is a ring
If (R, +, •) is a ring, prove that C = |r ∈ R|ar = ra, for all a ∈ R] is a subring of R. (The subring C is called the center of R.)
Given a finite field F, let M2(F) denote the set of all 2 Ã 2 matrices with entries from F. As in Example 14.2, (M2(F), +, ¢) becomes a noncommutative ring with unity.a)
Given an integral domain (D, +, •) with zero element z, let a, b ∈ D with ab ≠ z. (a) If a3 = b3 and a5 = b5, prove that a = b. (b) Let m, n ∈ Z+ with gcd(m, n) = 1. If am = bm and an = bn,
Let A = R+. Define ⊕ and ⊙ on A by a ⊕ b = ab, the ordinary product of a, b; and a ⊙ b = alog2b.(a) Verify that (A, ⊕, ⊙) is a commutative ring with unity.(b) Is this ring an integral
Let R be a ring with ideals A and B. Define A + B = [a + b|a ∈ A, b ∈ B}. Prove that A + F is an ideal of R. (For any ring R, the ideals of R form a poset under set inclusion. If A and B are
If x, y, and z are Boolean variables and x + y + z = xyz, prove that x, y, z all have the same value.
Simplify the following Boolean expressions.(a) xy + (x + y) + y(c) yz + wx + z + [wz(xy + wz)]
Find the values of the Boolean variables w, x, y, z that satisfy the following system of simultaneous (Boolean) equations. x + y = 0 y = z y + + zw = w
(a) For f, g, h: Bn → B, prove that fg + h + gh = fg + h and that fg + f + g + = 1. (b) State the dual of each result in part (a).
Let f, g: Bn → B. Define the relation "
Define the closed binary operation Š• (Exclusive or) on Fn, the set of all Boolean functions on n variables, by f Š• g = f + g, where f, g: Bn †’ B.(a) Determine f Š• f, f Š• , f
Let if, x, and y be Boolean variables where the value of x is 1. For each of the following Boolean expressions, determine, if possible, the value of the expression. If you cannot determine the value
(a) Find the fundamental conjunction made up from the variables w, x, y, z, or their complements, where the value of the conjunction is 1 precisely when (i) w = x = 0, y = z = 1. (ii) w = 0, x = 1, y
Suppose that f: B3 B is defined by(a) Determine the d.n.f. and c.n.f. for f. (b) Write f as a sum of minterms and as a product of maxterms (utilizing binary labels).
Let g: B4 → B be defined by g(w, x, y, z) = (wz + xyz)(x + z). (a) Find the d.n.f. and c.n.f. for g. (b) Write g as a sum of minterms and as a product of max- terms (utilizing binary labels).
Let f: B4 → B. Find the disjunctive normal form for f if (a) f-1(1) = {0101 (that is, w = 0, x = 1, y = 0, z = 1), 0110, 1000, 1011}. (b) f-10) = {0000, 0001, 0010, 0100, 1000, 1001, 0110}.
Using inverters, AND gates, and OR gates, construct the gates shown in Fig. 15.6.
Obtain a minimal-product-of-sums representation for f(w, x, y, z) = Π M(0, 1, 2, 4, 5, 10, 12, 13, 14).
In each of the following, f: B4 → B, where the Boolean variables (in order) are w, x, y, and z. Determine |f-1(0)| and |f-1 (1) if, as a minimal sum of products, f reduces to (a) (b) wy (c)
Using only NAND1 gates (see Fig. 15.6), construct the inverter, AND gate, and OR gate.
Answer Exercise 2, replacing NAND by NOR.Exercise 2:Using only NAND1 gates (see Fig. 15.6), construct the inverter, AND gate, and OR gate.
Using inverters, AND gates, and OR gates, construct gating networks for(a) f(x, y, z) = x + y + x(b) g(x, y, z) = (x + z)(y + )
Implement the half-adder of Fig. 15.3 using only(a) NAND gates;(b) NOR gates.
For each of the networks in Fig. 15.8 express the output in terms of the Boolean variables x, y or their complements. Then use the expression for the output to simplify the given network.
For each of the following Boolean functions f, design a two-level gating network for f as a minimal sum of products. (a) f: B3 → B, where f(x, y, z) = 1 if and only if exactly two of the variables
Find a minimal-sum-of-products representation for(a) f(w, x, y) = ∑, m(1, 2, 5, 6)(b) f(w, x, y) = Π[ M(0, 1,4, 5)(c) f(w, x, y, z) = ∑ m(0, 2, 5, 7, 8, 10, 13, 15)(d) f(w, x, y, z) = ∑ m{5,
For his tenth birthday, Mona wants to buy her son Jason some stamps for his collection. At the hobby shop she finds six different packages (which we shall call u, v, w, x, y, z). The kinds of stamps
Rework Example 15.20 using a Karnaugh map on six variables.Example 15.20For the graph shown in Fig. 15.9, let the vertices represent cities and the edges highways. We wish to build hospitals in some
Find a minimal-sum-of-products representation for (a) f(w, x, y, z) = ∑m(1, 3, 5, 7, 9) + d(10, 11, 12, 13, 14, 15) (b) f(w, x, y, z) = ∑ m(0, 5, 6, 8, 13, 14) + d(4, 9, 11) (c) f(v, w, x, y, z)
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
Determine all minimal dominating sets for the graph G shown in Fig. 15.13.
Verify the second distributive law and the identity and inverse laws for Example 15.25. Example 15.25 Let B be the set of all positive integer divisors of 30: B = {1, 2, 3, 5, 6, 10, 15, 30}. For all
If B is a Boolean algebra, prove that the zero elements and the one element of B are unique.
Let f: B1 → B2 be an isomorphism of Boolean algebras. Prove each of the following: (a) f(0) = 0. (b) f(1) = 1. (c) If x, y ∈ B1 with x ≤ y, then in B2, f(x) ≤ f(y). (d) If x is an atom of B1,
Let B1 be the Boolean algebra of all positive integer divisors of 2310, with B2 the Boolean algebra of all subsets of {a, b, c, d, e}. (a) Define f: B1 → B2 so that f(2) = {a}, f(3) = {b}, f(5) =
(b) State and prove another result comparable to that in part (a). (What principle is used here?)
Prove that the function f in Theorem 15.9 is one-to-one and onto. Theorem 15.9: Every finite Boolean algebra B is isomorphic to a Boolean algebra of sets.
Let B be a finite Boolean algebra with the n atoms x1, x2,..., xn. (So |B| = 2n.) Prove that 1 = x1 + x2 + ... + xn.
Complete the proof of Theorem 15.3.Theorem 15.3:(a) x 0 = 0(a)ʹ x + 1 = 1 Dominance Laws(b) x(x + y) = x bf x xy = x Absorption Laws(c) [xy = xz and y = z] y = z
Let B be the set of positive integer divisors of 210, and define +, , and ¯ for B by x + y = lcm(x, y), x y = xy = gcd(x, y), and = 210/x. Determine each of the
For a Boolean algebra R the relation "≤" on B, defined by x ≤ y if xy = x, was shown to be a partial order. Prove that: (a) If x ≤ y then x + y = y; and (b) If x ≤ y then ≤ .
Let (B, +, ∙, ¯, 0, 1) be a Boolean algebra that is partially ordered by ≤. (a) If w ∈ B and w ≤ 0, prove that w = 0. (b) If x ∈ B and 1 ≤ x, prove that x = 1. (c) If y, z ∈ B with y
Let (B, +, ∙, ¯, 0, 1) be a Boolean algebra that is partially ordered by ≤. If w, x, y, z ∈ B with w ≤ x and y ≤ z, prove that (a) wy ≤ xy; and (b) w + y ≤ x + z.
If B is a Boolean algebra, partially ordered by ≤, and x, y ∈ B, what is the dual of the statement "x ≤ y"?
Let n ¥ 2. If xl is a Boolean variable for all 1 ¤ i ¤ n, prove that
The four input lines for the network in Fig. 15.15 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) Find
For (a) n = 60, and (b) n = 120, Explain why the positive integer divisors of n do not yield a Boolean algebra. (Here x + y = lcm(x, y), xy = gcd(x, y), = n/x, 1 is the zero element, and n is the
Let a, b, c ∈ B, a Boolean algebra. Prove that ab + c = a(b + c) if and only if c ≤ a.
Eileen is having a party and finds herself confronted with decisions about inviting five of her friends. (a) If she invites Margaret, she must also invite Joan. (b) If Kathleen is invited, Nettie and
Let B be a Boolean algebra that is partially ordered by ≤. If x, y, z ∈ B, prove that x + y ≤ z if and only if x ≤ z and y ≤ z.
State and prove the dual of the result in Exercise 5. Exercise 5: Let B be a Boolean algebra that is partially ordered by ≤. If x, y, z ∈ B, prove that x + y ≤ z if and only if x ≤ z and y
Let B be a Boolean algebra that is partially ordered by ≤. For all x, y ∈ B prove that (a) x ≤ y if and only if + y = 1; and (b) x ≤ if and only if xy = 0.
Let x, y be elements in the Boolean algebra B. Prove that x = y if and only if x + y = 0.
Use a Karnaugh map to find a minimal-sum-of products representation for (a) f(w, x, y, z) = ∑ m(0, 2, 3, 6, 7, 14, 15) (b) g(v, w, x, y, z) = Π M(1, 2, 4, 6, 9, 10, 11, 14, 17, 18, 19,20, 22,
For each of the following sets, determine whether or not the set is a group under the stated binary operation. If so, determine its identity and the inverse of each of its elements. If it is not a
Prove that a group G is abelian if and only if for alia, b ∈ G, (ab)-1 = a-1b-1.
Find all subgroups in each of the following groups, (a) (Z12, +) (b) (Z*11, ∙) (c) S3
(a) How many rigid motions (in two or three dimensions) are there for a square?(b) Make a group table for these rigid motions like the one in Table 16.5 for the equilateral triangle. What is the
(a) How many rigid motions (in two or three dimensions) are there for a regular pentagon? Describe them geometrically. (b) Answer part (a) for a regular n-gon, n ≥ 3.
In the group S5, letDetermine αβ, βα, α3, β4, α-1, β-1, (αβ)-1,
If G is a group, let H = [a ∈ G|ag = ga for all g ∈ G}. Prove that H is a subgroup of G. (The subgroup H is called the center of G.)
Let w be the complex number (1/√2) (1 + i). (a) Show that w8 = 1 but of wn ≠ 1 for n ∈ Z+, 1 ≤ n ≤ 7. (b) Verify that [wn|n ∈ Z+, 1 ≤ n ≤ 8] is an abelian group under multiplication.
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