## 数学代写|MAT270 discrete mathematics

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## 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）

- 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 $m=p \cdot n^2$.

We will prove both statements separately.

(1) Let $n = p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ be the prime factorization of $n$. Then $n^2 = p_1^{2a_1}p_2^{2a_2}\cdots p_k^{2a_k}$, which shows that the exponent of $p$ in the prime factorization of $n^2$ is even (or zero) if $p$ is not one of the $p_i$’s, and it is twice the exponent of $p$ in the prime factorization of $n$ if $p$ is one of the $p_i$’s. In either case, the exponent of $p$ in the prime factorization of $n^2$ is even (or zero), which proves (1).

(2) Let $m = p \cdot n^2$. We need to count the number of times $p$ appears in the prime factorization of $m$. By part (1), $p$ appears an even number (or zero) of times in the prime factorization of $n^2$. Therefore, $p$ appears an odd number of times in the prime factorization of $m$ if and only if it appears an odd number of times in the prime factorization of $p$. But $p$ is prime, so it appears in the prime factorization of $p$ exactly once. Therefore, $p$ appears an odd number of times in the prime factorization of $m$, which proves (2).

- Prove the following statements.

(1) There are no numbers $a, b$ such that $\frac{a}{b}=\sqrt{7}$.

(2) There are no numbers $a, b$ such that $\frac{a}{b}=\sqrt{23}$.

(3) Let $p$ be a prime. There are no numbers $a, b$ such that $\frac{a}{b}=\sqrt{p}$.

We will prove all three statements using a proof by contradiction.

(1) Suppose, for contradiction, that there exist integers $a$ and $b$ such that $\frac{a}{b}=\sqrt{7}$. Then $a^2=7b^2$. This implies that $a^2$ is a multiple of $7$, so $a$ must also be a multiple of $7$. Let $a=7c$. Then $49c^2=7b^2$, which simplifies to $7c^2=b^2$. This implies that $b$ is also a multiple of $7$. But this contradicts the assumption that $a$ and $b$ have no common factors, since they are both multiples of $7$. Therefore, there are no integers $a$ and $b$ such that $\frac{a}{b}=\sqrt{7}$.

(2) Suppose, for contradiction, that there exist integers $a$ and $b$ such that $\frac{a}{b}=\sqrt{23}$. Then $a^2=23b^2$. This implies that $a^2$ is a multiple of $23$, so $a$ must also be a multiple of $23$. Let $a=23c$. Then $529c^2=23b^2$, which simplifies to $23c^2=b^2$. This implies that $b$ is also a multiple of $23$. But this contradicts the assumption that $a$ and $b$ have no common factors, since they are both multiples of $23$. Therefore, there are no integers $a$ and $b$ such that $\frac{a}{b}=\sqrt{23}$.

(3) Suppose, for contradiction, that there exist integers $a$ and $b$ such that $\frac{a}{b}=\sqrt{p}$ for some prime $p$. Then $a^2=pb^2$. This implies that $a^2$ is a multiple of $p$, so $a$ must also be a multiple of $p$. Let $a=pc$. Then $p^2c^2=pb^2$, which simplifies to $pc^2=b^2$. This implies that $b$ is also a multiple of $p$. But this contradicts the assumption that $a$ and $b$ have no common factors, since they are both multiples of $p$. Therefore, there are no integers $a$ and $b$ such that $\frac{a}{b}=\sqrt{p}$ for any prime $p$.

## 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.

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