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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 $n \in \mathbb{N}$. Show that the following statements are true
(1) $n^5 \bmod 5=n \bmod 5$.
(2) $n^7 \bmod 7=n \bmod 7$.
Proof:
(1) We can use Fermat’s Little Theorem, which states that if $p$ is a prime number and $a$ is an integer not divisible by $p$, then $a^{p-1}\equiv 1 \pmod p$. Applying this theorem with $p=5$ and $a=n$, we have $n^4\equiv 1 \pmod 5$. Multiplying both sides by $n$ gives $n^5 \equiv n \pmod 5$, which is the desired result.
(2) Similarly, we can use Fermat’s Little Theorem with $p=7$ and $a=n$, which gives $n^6\equiv 1 \pmod 7$. Multiplying both sides by $n$ gives $n^7 \equiv n \pmod 7$, which is the desired result.
Therefore, both statements are true.
- Prove the following statements. Let $n \in \mathbb{N}, n>1$, and $a \in \mathbb{N}$
(1) If $a \bmod n=b \bmod n$, then $(-a) \bmod n=(-b) \bmod b$.
(2) If $p$ is odd prime, then $(-a)^p=-a^p$.
Proof:
(1) Since $a \bmod n=b \bmod n$, we have $n$ divides $a-b$. This means that there exists an integer $k$ such that $a-b=kn$.
Now consider $(-a) \bmod n$ and $(-b) \bmod n$. We want to show that they are equal. We have:
$(-a) \bmod n=(-a+kn-kn) \bmod n=(-a+kn) \bmod n=-(a-kn) \bmod n=-(b-kn) \bmod n=(-b+kn) \bmod n=(-b) \bmod n$
where we have used the fact that adding or subtracting a multiple of $n$ does not change the remainder modulo $n$. Therefore, we have shown that $(-a) \bmod n=(-b) \bmod n$.
(2) We can use the binomial theorem to expand $(-a)^p$:
$(-a)^p=(-1)^p a^p$
If $p$ is odd, then $(-1)^p=-1$, so we have $(-a)^p=-a^p$, which is the desired result.
Therefore, both statements 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.
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