MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH1004 Discrete Mathematics离散数学课程的代写代考和辅导服务！
这是悉尼大学 离散数学课程的代写成功案例。
MATH1004课程简介
This unit provides an introduction to fundamental aspects of discrete mathematics, which deals with ‘things that come in chunks that can be counted’. It focuses on the enumeration of a set of numbers, viz. Catalan numbers. Topics include sets and functions, counting principles, discrete probability, Boolean expressions, mathematical induction, linear recurrence relations, graphs and trees.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. identify combinatorial objects involved in counting problems
- LO2. understand how to construct switching circuits representing Boolean functions
- LO3. factor numbers using sieve methods and use the Euclidean algorithm to compute greatest common divisors
- LO4. solve linear recurrence relations by using generating functions or characteristic equations.
MATH1004 Discrete Mathematics HELP（EXAM HELP， ONLINE TUTOR）
In $\mathbb{Z}_{12}$
(a) which elements have no multiplicative inverse?
(b) what is the multiplicative inverse of each of the remaining elements?
(c) Prove that the invertible elements form a group, and write out the multiplication table for this group.
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.
Given a $\operatorname{set} G$ with a binary operation (denoted by adjacency of elements) that is both closed and associative, suppose we have the following axiom in place of our usual identity axiom:
G3.’ Given $x \in G$, there is an element $e \in G$ (possibly depending on $x$ ) such that $x e=x$.
Decide whether or not the usual identity axiom holds.
Consider the set $\mathbb{Q}^{\times}$of non-zero rational numbers, and define the binary operation $*$ as follows:
$$
x * y=\left{\begin{aligned}
x y & , \text { if } x, y>0 \
x y & , \text { if } x y<0 \
2 x y & , \text { if } x, y<0
\end{aligned}\right.
$$
That is, if $x$ and $y$ are both positive, then $x * y$ is the usual product, and if one of $x$ and $y$ is positive, then $x * y$ is also the usual product, but if both $x$ and $y$ are negative, then $x * y$ is twice the usual product.
Verify the four group axioms for $\left(\mathbb{Q}^{\times}, *\right)$.
(6 6 ths ea) Prove the following:
(a) The product of two rational numbers is a rational number.
(b) The average of two rational numbers is also rational.
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