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这是昆士兰大学离散数学课程的代写成功案例。
MATH1061课程简介
Course Description: Propositional & predicate logic, valid arguments, methods of proof. Elementary set theory. Elementary graph theory. Relations & functions. Induction & recursive definitions. Counting methods pigeonhole, inclusion/exclusion. Introductory probability. Binary operations, groups, fields. Applications of finite fields. Elementary number theory.
There is no particular assumed background, apart from a level of mathematical sophistication roughly equivalent to completion of Queensland Mathematical Methods (formerly Maths B) at high school or MATH1040. Concurrent enrolment in MATH1061 and MATH1040 is appropriate for students who do not have Mathematical Methods from high school but who have a strong mathematical background, however this will require you to do extra work in MATH1061.
Prerequisites
This course provides an introduction to discrete mathematics. It is likely to be useful for students who are planning on studying more mathematics, those intending to teach, and also for those enrolled in computer science, engineering, science and information technology.
Course Changes in Response to Previous Student Feedback
This is an active learning course. Learning resources have been updated and extended.
MATH1061 Discrete Mathematics HELP（EXAM HELP， ONLINE TUTOR）
One labeling of the regular hexagon is the figure below, where any symmetry operation on the hexagon permutes the sets of elements that share a number. For example, the three edges numbered 4 may be interchanged with the three edges numbered 5 .
(a) Write the five permutations from $S_5$ that represent non-trivial rotations about an axis perpendicular to the plane of the hexagon.
(b) Write the three permutations from $S_5$ that represent reflections about axes between two opposite vertices (or, if you prefer, $180^{\circ}$ rotations about theses axes).
(c) Express the last three non-trivial symmetries of the regular hexagon as permutations from $S_5$, and describe what each does geometrically.
Consider the following six functions:
$$
\begin{aligned}
f_1(x) & =x & f_2(x) & =1-x \
f_3(x) & =\frac{1}{x} & f_4(x) & =\frac{1}{1-x} \
f_5(x) & =\frac{x}{x-1} & f_6(x) & =\frac{x-1}{x}
\end{aligned}
$$
Now think of these as a group under the operation composition. For example,
$$
\begin{aligned}
\left(f_2 \circ f_3\right)(x) & =f_2\left(f_3(x)\right) \
& =f_2\left(\frac{1}{x}\right) \
& =1-\frac{1}{x} \
& =\frac{x-1}{x} \
& =f_6(x) .
\end{aligned}
$$
What group is this?
In $\mathbb{Z}_{13}$
(a) what is the additive inverse of 5
(b) for what $m$ and $n$ does $13 m+5 n=1$ ?
(c) what is the multiplicative inverse of 5
(d) what is the square root of -1
Given a set $G$ with a binary operation (denoted by adjacency of elements) that satisfies the three following axioms:
G1′. Given any $x, y, z \in G,(x y) z=x(y z)$.
G2′. Given any $a, b \in G$, there is a unique element $x \in G$ such that $x a=b$.
G3′. Given any $a, b \in G$, there is a unique element $y \in G$ such that $a y=b$
Show that $G$ is a group under this operation.
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