Discrete Mathematics
5th Edition
ISBN: 9780134689562
Author: Dossey, John A.
Publisher: Pearson,
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Chapter 3, Problem 9SE
To determine
To perform: The operation
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Task Description:
Read the following case study and answer the questions that follow.
Ella is a 9-year-old third-grade student in an inclusive classroom. She has been diagnosed with Emotional and Behavioural Disorder (EBD). She has been struggling academically and socially due to
challenges related to self-regulation, impulsivity, and emotional outbursts. Ella's behaviour includes frequent tantrums, defiance toward authority figures, and difficulty forming positive relationships with peers. Despite her challenges, Ella shows an interest in art and creative activities and demonstrates strong verbal skills when calm.
Describe 2 strategies that could be implemented that could help Ella regulate her emotions in class (4 marks)
Explain 2 strategies that could improve Ella’s social skills (4 marks)
Identify 2 accommodations that could be implemented to support Ella academic progress and provide a rationale for your recommendation.(6 marks)
Provide a detailed explanation of 2 ways…
1. Iodine-131 is tone of the most commonly used radioactive isotopes of iodine. It is used to treat hyper-
thyroidism and some kinds of thyroid cancer.
(a) Iodine-131 has a half-life of about 8 days. Find an expression for I(t), the mass of Iodine-131
remaining after t days, in terms of t and Io, the initial mass of Iodine-131 present at time t = 0.
(b) If a dose of 0.9 mg of Iodine-131 is administered, how much is still present after 24 hours?
(c) How much Iodine-131 is present after one week? Does your answer make sense?
Question 2: When John started his first job, his first end-of-year salary was $82,500. In the following years, he received salary raises as shown in the following table.
Fill the Table: Fill the following table showing his end-of-year salary for each year. I have already provided the end-of-year salaries for the first three years. Calculate the end-of-year salaries for the remaining years using Excel. (If you Excel answer for the top 3 cells is not the same as the one in the following table, your formula / approach is incorrect) (2 points)
Geometric Mean of Salary Raises: Calculate the geometric mean of the salary raises using the percentage figures provided in the second column named “% Raise”. (The geometric mean for this calculation should be nearly identical to the arithmetic mean. If your answer deviates significantly from the mean, it's likely incorrect. 2 points)
Starting salary
% Raise
Raise
Salary after raise
75000
10%
7500
82500
82500
4%
3300…
Chapter 3 Solutions
Discrete Mathematics
Ch. 3.1 - In Exercises 1–8, find the quotient and remainder...Ch. 3.1 - Prob. 2ECh. 3.1 - Prob. 3ECh. 3.1 - In Exercises 1–8, find the quotient and remainder...Ch. 3.1 - In Exercises 1–8, find the quotient and remainder...Ch. 3.1 - Prob. 6ECh. 3.1 - Prob. 7ECh. 3.1 - Prob. 8ECh. 3.1 - Prob. 9ECh. 3.1 - Prob. 10E
Ch. 3.1 - Prob. 11ECh. 3.1 - Prob. 12ECh. 3.1 - Prob. 13ECh. 3.1 - In Exercises 9–16, determine whether p ≡ q (mod...Ch. 3.1 - Prob. 15ECh. 3.1 - Prob. 16ECh. 3.1 - Prob. 17ECh. 3.1 - Prob. 18ECh. 3.1 - Prob. 19ECh. 3.1 - In Exercises 17–36, perform the indicated...Ch. 3.1 - Prob. 21ECh. 3.1 - Prob. 22ECh. 3.1 - Prob. 23ECh. 3.1 - Prob. 24ECh. 3.1 - Prob. 25ECh. 3.1 - Prob. 26ECh. 3.1 - Prob. 27ECh. 3.1 - In Exercises 17–36, perform the indicated...Ch. 3.1 - Prob. 29ECh. 3.1 - Prob. 30ECh. 3.1 - Prob. 31ECh. 3.1 - Prob. 32ECh. 3.1 - Prob. 33ECh. 3.1 - In Exercises 17–36, perform the indicated...Ch. 3.1 - Prob. 35ECh. 3.1 - Prob. 36ECh. 3.1 - Prob. 37ECh. 3.1 - A hospital heart monitoring device uses two feet...Ch. 3.1 - Prob. 39ECh. 3.1 - Use Example 3.2 to determine the correct check...Ch. 3.1 - Prob. 41ECh. 3.1 - Federal Express packages carry a 10-digit...Ch. 3.1 - Prob. 43ECh. 3.1 - Use the formula in Example 3.7 to determine all...Ch. 3.1 - Let A denote the equivalence class containing 4 in...Ch. 3.1 - Prob. 46ECh. 3.1 - Let R be the equivalence relation defined in...Ch. 3.1 - Show that there exist integers m, x, and y such...Ch. 3.1 - Prob. 49ECh. 3.1 - A project has the nine tasks T1, T2, T3, T4, T5,...Ch. 3.1 - Prob. 51ECh. 3.1 - Prob. 52ECh. 3.2 - List, in increasing order, the divisors of 45
Ch. 3.2 - Prob. 2ECh. 3.2 - Prob. 3ECh. 3.2 - List, in increasing order, the common divisors of...Ch. 3.2 - Prob. 5ECh. 3.2 - Prob. 6ECh. 3.2 - Prob. 7ECh. 3.2 - Prob. 8ECh. 3.2 - Prob. 9ECh. 3.2 - In Exercises 5–10, make a table such as the one...Ch. 3.2 - Prob. 11ECh. 3.2 - Prob. 12ECh. 3.2 - Prob. 13ECh. 3.2 - Prob. 14ECh. 3.2 - Prob. 15ECh. 3.2 - Prob. 16ECh. 3.2 - Prob. 17ECh. 3.2 - In Exercises 13–18, make a table such as the one...Ch. 3.2 - Prob. 19ECh. 3.2 - In Exercises 19–22, use the Euclidean algorithm to...Ch. 3.2 - Prob. 21ECh. 3.2 - Prob. 22ECh. 3.2 - Prob. 23ECh. 3.2 - Prob. 24ECh. 3.2 - Prob. 25ECh. 3.2 - In Exercises 23–26, use the extended Euclidean...Ch. 3.2 - Prob. 27ECh. 3.2 - Prob. 28ECh. 3.2 - Prob. 29ECh. 3.2 - Prob. 30ECh. 3.2 - Prob. 31ECh. 3.2 - Prob. 32ECh. 3.2 - Prob. 33ECh. 3.3 - Prob. 1ECh. 3.3 - In Exercises 1–4, change the given plaintext...Ch. 3.3 - Prob. 3ECh. 3.3 - In Exercises 1–4, change the given plaintext...Ch. 3.3 - Prob. 5ECh. 3.3 - In Exercises 5–10, apply the modular...Ch. 3.3 - Prob. 7ECh. 3.3 - In Exercises 5–10, apply the modular...Ch. 3.3 - Prob. 9ECh. 3.3 - Prob. 10ECh. 3.3 - Prob. 11ECh. 3.3 - In Exercises 11–14, find b corresponding to the...Ch. 3.3 - Prob. 13ECh. 3.3 - Prob. 14ECh. 3.3 - Prob. 15ECh. 3.3 - Prob. 16ECh. 3.3 - Prob. 17ECh. 3.3 - Prob. 18ECh. 3.3 - Prob. 19ECh. 3.3 - In Exercises 15–22, use the extended Euclidean...Ch. 3.3 - Prob. 21ECh. 3.3 - Prob. 22ECh. 3.3 - Prob. 23ECh. 3.3 - Suppose n = 93, E = 17, and the ciphertext message...Ch. 3.3 - Prob. 25ECh. 3.3 - Prob. 26ECh. 3.3 - Prob. 27ECh. 3.3 - Prob. 28ECh. 3.3 - Prob. 29ECh. 3.3 - Prob. 30ECh. 3.3 - Prob. 31ECh. 3.3 - Prob. 32ECh. 3.3 - Prob. 33ECh. 3.3 - Prob. 34ECh. 3.3 - Prob. 35ECh. 3.3 - Prob. 36ECh. 3.3 - Prob. 37ECh. 3.4 - In Exercises 1–8, determine the parity check digit...Ch. 3.4 - In Exercises 1–8, determine the parity check digit...Ch. 3.4 - Prob. 3ECh. 3.4 - Prob. 4ECh. 3.4 - Prob. 5ECh. 3.4 - Prob. 6ECh. 3.4 - Prob. 7ECh. 3.4 - In Exercises 1–8, determine the parity check digit...Ch. 3.4 - Prob. 9ECh. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - In Exercises 9–16, use formula (3.1) to determine...Ch. 3.4 - Prob. 13ECh. 3.4 - In Exercises 9–16, use formula (3.1) to determine...Ch. 3.4 - Prob. 15ECh. 3.4 - Prob. 16ECh. 3.4 - Prob. 17ECh. 3.4 - In Exercises 17–24, determine the Hamming distance...Ch. 3.4 - Prob. 19ECh. 3.4 - Prob. 20ECh. 3.4 - Prob. 21ECh. 3.4 - Prob. 22ECh. 3.4 - Prob. 23ECh. 3.4 - Prob. 24ECh. 3.4 - Prob. 25ECh. 3.4 - Prob. 26ECh. 3.4 - Prob. 27ECh. 3.4 - Prob. 28ECh. 3.4 - Prob. 29ECh. 3.4 - Prob. 30ECh. 3.4 - Prob. 31ECh. 3.4 - In Exercises 25–32, add the given codewords using...Ch. 3.4 - In Exercises 33–36, suppose that the minimal...Ch. 3.4 - In Exercises 33–36, suppose that the minimal...Ch. 3.4 - In Exercises 33–36, suppose that the minimal...Ch. 3.4 - Prob. 36ECh. 3.4 - Prob. 37ECh. 3.4 - Prob. 38ECh. 3.4 - Prob. 39ECh. 3.4 - Prob. 41ECh. 3.5 - Prob. 1ECh. 3.5 - In Exercises 1–4, determine the number of words in...Ch. 3.5 - Prob. 3ECh. 3.5 - Prob. 4ECh. 3.5 - Prob. 5ECh. 3.5 - Prob. 6ECh. 3.5 - Prob. 7ECh. 3.5 - In Exercises 5–8, suppose that the generator...Ch. 3.5 - Prob. 9ECh. 3.5 - Prob. 10ECh. 3.5 - In Exercises 9–12, determine the size of the check...Ch. 3.5 - In Exercises 9–12, determine the size of the check...Ch. 3.5 - If the check matrix of a matrix code is a 9 × 3...Ch. 3.5 - If the check matrix of a matrix code is an 11 × 4...Ch. 3.5 - Prob. 15ECh. 3.5 - In Exercises 15–20, determine all the codewords...Ch. 3.5 - Prob. 17ECh. 3.5 - Prob. 18ECh. 3.5 - Prob. 19ECh. 3.5 - Prob. 20ECh. 3.5 - Prob. 21ECh. 3.5 - Prob. 22ECh. 3.5 - Prob. 23ECh. 3.5 - In Exercises 21–28, determine the check matrix...Ch. 3.5 - Prob. 25ECh. 3.5 - Prob. 26ECh. 3.5 - Prob. 27ECh. 3.5 - Prob. 28ECh. 3.5 - Prob. 29ECh. 3.5 - Exercises 29 and 30, the check matrix A* for a...Ch. 3.5 - Prob. 31ECh. 3.5 - In Exercises 31–38, use Theorem 3.8(b) to...Ch. 3.5 - Prob. 33ECh. 3.5 - Prob. 34ECh. 3.5 - Prob. 35ECh. 3.5 - Prob. 36ECh. 3.5 - Prob. 37ECh. 3.5 - Prob. 38ECh. 3.5 - Prob. 39ECh. 3.5 - Consider the (3, 7)-code with generator...Ch. 3.5 - Prob. 41ECh. 3.5 - Find the generator matrix of the code that encodes...Ch. 3.5 - Prob. 43ECh. 3.6 - Prob. 1ECh. 3.6 - Prob. 2ECh. 3.6 - Prob. 3ECh. 3.6 - Prob. 4ECh. 3.6 - In Exercises 1–8, determine the syndrome of each...Ch. 3.6 - In Exercises 1–8, determine the syndrome of each...Ch. 3.6 - Prob. 7ECh. 3.6 - Prob. 8ECh. 3.6 - Prob. 9ECh. 3.6 - Prob. 10ECh. 3.6 - Prob. 11ECh. 3.6 - Prob. 12ECh. 3.6 - Prob. 13ECh. 3.6 - Prob. 14ECh. 3.6 - Prob. 15ECh. 3.6 - Prob. 16ECh. 3.6 - Prob. 17ECh. 3.6 - Prob. 18ECh. 3.6 - Prob. 19ECh. 3.6 - In Exercises 9–28, the given word was received...Ch. 3.6 - Prob. 21ECh. 3.6 - In Exercises 9–28, the given word was received...Ch. 3.6 - In Exercises 9–28, the given word was received...Ch. 3.6 - Prob. 24ECh. 3.6 - Prob. 25ECh. 3.6 - Prob. 26ECh. 3.6 - Prob. 27ECh. 3.6 - Prob. 28ECh. 3.6 - Prob. 29ECh. 3.6 - In Exercises 29 and 30, a check matrix and a list...Ch. 3.6 - Prob. 31ECh. 3.6 - Prob. 32ECh. 3.6 - Prob. 33ECh. 3.6 - In Exercises 31–34, determine the minimal value of...Ch. 3.6 - Prob. 35ECh. 3.6 - Prob. 36ECh. 3.6 - Prob. 37ECh. 3.6 - In Exercises 35–38, determine the smallest values...Ch. 3.6 - Prob. 39ECh. 3.6 - Prob. 40ECh. 3.6 - Prob. 41ECh. 3.6 - Prove by mathematical induction that r2 + 1 ≤ 2r...Ch. 3 - Prob. 1SECh. 3 - Prob. 2SECh. 3 - Determine whether each statement in Exercises 1–4...Ch. 3 - Prob. 4SECh. 3 - Prob. 5SECh. 3 - Prob. 6SECh. 3 - Prob. 7SECh. 3 - Prob. 8SECh. 3 - Prob. 9SECh. 3 - Prob. 10SECh. 3 - Prob. 11SECh. 3 - Prob. 12SECh. 3 - Prob. 13SECh. 3 - Prob. 14SECh. 3 - Prob. 15SECh. 3 - Prob. 16SECh. 3 - Prob. 17SECh. 3 - Prob. 18SECh. 3 - Prob. 19SECh. 3 - Prob. 20SECh. 3 - Prob. 21SECh. 3 - Prob. 22SECh. 3 - Prob. 23SECh. 3 - Prob. 24SECh. 3 - Prob. 25SECh. 3 - Prob. 26SECh. 3 - Prob. 27SECh. 3 - Prob. 28SECh. 3 - Prob. 29SECh. 3 - Prob. 30SECh. 3 - Prob. 31SECh. 3 - Prob. 32SECh. 3 - Prob. 33SECh. 3 - Prob. 34SECh. 3 - Prob. 35SECh. 3 - Prob. 36SECh. 3 - Prob. 37SECh. 3 - Prob. 38SECh. 3 - Prob. 39SECh. 3 - Prob. 40SECh. 3 - Prob. 41SECh. 3 - Prob. 42SECh. 3 - Prob. 43SECh. 3 - Prob. 44SECh. 3 - Prob. 45SECh. 3 - Prob. 46SECh. 3 - Prob. 47SECh. 3 - Prob. 48SECh. 3 - Prob. 49SECh. 3 - Prob. 50SECh. 3 - Prob. 51SECh. 3 - Prob. 52SECh. 3 - Prob. 53SECh. 3 - Prob. 54SECh. 3 - Prob. 55SECh. 3 - Prob. 56SECh. 3 - Prob. 57SECh. 3 - Prob. 58SECh. 3 - Prob. 59SECh. 3 - Prob. 60SECh. 3 - Prob. 62SECh. 3 - Prob. 63SECh. 3 - Prob. 64SECh. 3 - Prob. 65SECh. 3 - Prob. 66SECh. 3 - Prob. 67SECh. 3 - Prob. 68SECh. 3 - Prob. 69SECh. 3 - Prob. 70SECh. 3 - Prob. 71SECh. 3 - Prob. 72SECh. 3 - Prob. 73SECh. 3 - Prob. 2CPCh. 3 - Prob. 3CPCh. 3 - Prob. 4CPCh. 3 - Prob. 5CPCh. 3 - Prob. 6CPCh. 3 - Prob. 7CP
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- d₁ ≥ ≥ dn ≥ 0 with di even. di≤k(k − 1) + + min{k, di} vi=k+1 T2.5: Let d1, d2,...,d be integers such that n - 1 Prove the equivalence of the Erdos-Gallai conditions: for each k = 1, 2, ………, n and the Edge-Count Criterion: Σier di + Σjeл(n − 1 − d;) ≥ |I||J| for all I, JC [n] with In J = 0.arrow_forwardT2.4: Let d₁arrow_forwardSolve the following boundary value problem using method of separation of variables: 1 ə ди r dr 70% (107) + 1 д²и = 0, 12802 -πarrow_forwardT2.3: Prove that there exists a connected graph with degrees d₁ ≥ d₂ >> dn if and only if d1, d2,..., dn is graphic, d ≥ 1 and di≥2n2. That is, some graph having degree sequence with these conditions is connected. Hint - Do not attempt to directly prove this using Erdos-Gallai conditions. Instead work with a realization and show that 2-switches can be used to make a connected graph with the same degree sequence. Facts that can be useful: a component (i.e., connected) with n₁ vertices and at least n₁ edges has a cycle. Note also that a 2-switch using edges from different components of a forest will not necessarily reduce the number of components. Make sure that you justify that your proof has a 2-switch that does decrease the number of components.arrow_forwardT2.2 Prove that a sequence s d₁, d₂,..., dn with n ≥ 3 of integers with 1≤d; ≤ n − 1 is the degree sequence of a connected unicyclic graph (i.e., with exactly one cycle) of order n if and only if at most n-3 terms of s are 1 and Σ di = 2n. (i) Prove it by induction along the lines of the inductive proof for trees. There will be a special case to handle when no d₂ = 1. (ii) Prove it by making use of the caterpillar construction. You may use the fact that adding an edge between 2 non-adjacent vertices of a tree creates a unicylic graph.arrow_forwardI need help with this problem and an explanation of the solution for the image described below. (Statistics: Engineering Probabilities)arrow_forward= == T2.1: Prove that the necessary conditions for a degree sequence of a tree are sufficient by showing that if di 2n-2 there is a caterpillar with these degrees. Start the construction as follows: if d1, d2,...,d2 and d++1 = d = 1 construct a path v1, v2, ..., vt and add d; - 2 pendent edges to v, for j = 2,3,..., t₁, d₁ - 1 to v₁ and d₁ - 1 to v₁. Show that this construction results vj in a caterpillar with degrees d1, d2, ..., dnarrow_forwardDo the Laplace Transformation and give the answer in Partial Fractions. Also do the Inverted Laplace Transformation and explain step-by-step.arrow_forwardI need help with this problem and an explanation of the solution for the image described below. (Statistics: Engineering Probabilities)arrow_forward12. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.508.XP. ASK YOUR TEA Make a substitution to express the integrand as a rational function and then evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) x + 16 dx X Need Help? Read It SUBMIT ANSWER 13. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.512.XP. ASK YOUR TEA Make a substitution to express the integrand as a rational function and then evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) dx 8)(2x + 1) Need Help? Read It SUBMIT ANSWER 14. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.518.XP. Find the area of the region under the given curve from 1 to 5. y = x² +7 6x - x² Need Help? Read It ASK YOUR TEAarrow_forwardLakshmi planted 20 begonias, but her neighbor’s dog ate 7 of them. What percent of the begonias did the dog eat?arrow_forwardDETAILS MY NOTES SESSCALCET2 6.3.012. 6. [-/1 Points] Evaluate the integral. x-4 dx x² - 5x + 6 Need Help? Read It SUBMIT ANSWER 7. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.019. Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) x²+1 (x-6)(x-5)² dx Need Help? Read It SUBMIT ANSWER 8. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.3.021. Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) ✓ x² 4 +4 dxarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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