Statistics-lab™可以为您提供tesu.edu MAT270 discrete mathematics离散数学课程的代写代考和辅导服务！
MAT270 discrete mathematics课程简介
Discrete Mathematics is designed to meet the needs not only of students majoring in computer science but of a wider audience, especially students in mathematics and science. The course provides tools for formal reasoning. Topics include counting rules, propositional and first-order logic, set theory, functions (with an emphasis on recursive functions), partial order and equivalence relations, Boolean algebra, and switching circuits. Graphs and trees are also introduced. With an emphasis on communication skills, students are required to interpret, describe, discuss, and justify conclusions based on logical reasoning. While the particular focus of the course is on reasoning related to computer programs, no knowledge of programming is required.
Advisory: It is advisable to have knowledge in a course equivalent to MAT-121: College Algebra with a grade of C or better to succeed in this course. Students are responsible for making sure that they have the necessary knowledge.
PREREQUISITES
Discrete Mathematics is a fundamental course that provides a solid foundation for computer science, mathematics, and science students. The course covers a wide range of topics, including counting rules, logic, set theory, functions, relations, Boolean algebra, and graph theory. The emphasis on formal reasoning and communication skills makes this course valuable to all students.
To succeed in this course, it is advisable to have a good understanding of college algebra, as it forms the basis of many concepts covered in the course. Students should ensure that they have the necessary prerequisite knowledge before enrolling in the course.
The course provides tools for formal reasoning, which are essential for students pursuing computer science or related fields. These tools enable students to analyze and design algorithms, write correct programs, and prove the correctness of programs.
Overall, discrete mathematics is a challenging but rewarding course that prepares students for further study in computer science, mathematics, and science. The skills and knowledge gained in this course are valuable not only in academia but also in industry and other fields.
MAT270 discrete mathematics HELP（EXAM HELP， ONLINE TUTOR）
- At some point during the baseball season, the $n$ teams of the American League have already played several games. Suppose team $i$ has won $w_i$ games so far, and $g_{i j}=g_{j i}$ is the number of games that teams $i$ and $j$ have yet to play. No game ends in a tie, so each game yet to be played gives one point to one of the teams and zero point to the other. You would like to decide if your favorite team (Red Sox?), say team $n$, can still win. In other words, you would like to determine whether there exists an outcome of the games to be played (remember, with no ties) such that team $n$ has at least as many victories as any other team (we allow team $n$ to be tied for first place).
Your article should contain at least the following:
- some motivation indicating that the problem is not as trivial as it appears,
- a construction (with proof and explanations) that this problem can be cast and solved as a maximum flow problem,
- a theorem (with proof and explanations) that team $n$ can win if and only if there is no set $S \subseteq{1, \cdots, n-1}$ such that
$$
w_n+\sum_{i=1}^{n-1} g_{i n}<\frac{\sum_{i \in S} w_i+\sum_{i<j, i \in S, j \in S} g_{i j}}{|S|}
$$
Motivation: At first glance, this problem might seem like a straightforward comparison of the number of wins of team $n$ with the other teams’ wins. However, the fact that there are still games to be played, and that each game can result in a point for only one team, makes this problem more complicated. It is not immediately clear how to determine whether team $n$ can still win the season. We need a more systematic approach to solve this problem.
Construction: We can model this problem as a maximum flow problem on a network. Consider a network with a source vertex $s$, a sink vertex $t$, and $n$ intermediate vertices $v_1, v_2, \ldots, v_n$. We draw an edge from $s$ to each $v_i$ with capacity $w_n-w_i$, representing the number of wins that team $n$ needs to catch up with team $i$. Similarly, we draw an edge from each $v_i$ to $t$ with capacity $\infty$, representing the number of remaining games that team $i$ needs to play. Finally, we draw an edge from each $v_i$ to each $v_j$ with capacity $g_{ij}$, representing the number of remaining games between teams $i$ and $j$.
We claim that the maximum flow in this network is equal to the number of games that team $n$ needs to win in order to have at least as many victories as any other team. To see why, consider any flow in the network. The flow on the edge from $s$ to $v_i$ represents the number of wins that team $n$ obtains against team $i$, and the flow on the edge from $v_i$ to $t$ represents the number of wins that team $i$ obtains in the remaining games. Therefore, the total number of wins of team $n$ in this flow is $\sum_{i=1}^{n-1} (w_n-w_i) – f(s,v_i)$, where $f(s,v_i)$ is the flow on the edge from $s$ to $v_i$. Similarly, the total number of wins of team $i$ in this flow is $w_i+f(v_i,t)$. Since there are no ties, the maximum possible number of wins is the total number of remaining games, which is $\frac{1}{2}\sum_{i<j} g_{ij}$. Therefore, team $n$ can win if and only if there exists a flow in the network with value at least $\sum_{i=1}^{n-1} (w_n-w_i) + \frac{1}{2}\sum_{i<j} g_{ij}$.
Theorem: Team $n$ can win if and only if there is no set $S \subseteq {1,\ldots,n-1}$ such that
w_n + \sum_{i \in S} g_{in} < \frac{\sum_{i \in S} w_i + \sum_{i<j, i \in S, j \in S} g_{ij}}{|S|}.wn+i∈S∑gin<∣S∣∑i∈Swi+∑i<j,i∈S,j∈Sgij.
Proof: Suppose that there exists a set $S$ that satisfies the inequality above. Then, we can set the flow on the edges from $s$ to $v_i$ as follows:
f(s,v_i) = \begin{cases} w_n – w_i, & i \in S \\ w_n – w_i + g_{in} – \frac{\sum_{j \in S} g_{ij}}{|S|}, & i \notin S. \end{cases}f(s,vi)={wn−wi,wn−wi+gin−∣S∣∑j∈Sgij,i∈Si∈/S.
It is easy
- Let $n \in \mathbb{N}$, and let $p$ be a prime. Show that the following statements are true
(1) $p$ appears an even number (or zero) of times in the prime factorization of $n^2$.
(2) $p$ appears an odd number of times in the prime factorization of $m=p \cdot n^2$.
Proof: (1) We can write $n^2$ as a product of primes, i.e., $n^2 = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, where $p_1, p_2, \dots, p_k$ are distinct primes and $a_1, a_2, \dots, a_k$ are positive integers. Thus, the prime factorization of $n^2$ contains each prime factor $p_i$ twice. Hence, $p$ appears an even number (or zero) of times in the prime factorization of $n^2$.
(2) We can write $m$ as $m = p \cdot n^2 = p \cdot p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$. Note that $p$ appears once in the prime factorization of $m$, and each $p_i$ appears an even number of times. Thus, $p$ appears an odd number of times in the prime factorization of $m$.
To see this, observe that for each $p_i$, there are two cases: either $p_i$ divides $p$ or it does not. If $p_i$ divides $p$, then $p_i$ appears once in the prime factorization of $m$. If $p_i$ does not divide $p$, then $p_i$ appears twice in the prime factorization of $n^2$, and hence $p_i^{a_i}$ appears an even number of times in the prime factorization of $m$. Therefore, the total number of times $p$ appears in the prime factorization of $m$ is odd.
Hence, the statements (1) and (2) are true.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
Statistics-lab™可以为您提供tesu.edu MAT270 discrete mathematics离散数学课程的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。