这个作业是来自美国的需要完成离散数学相关测试的数学代写，以下是具体作业内容：
CIT 592 Spring 2020 Homework 1
1. [10分]杰克和吉尔想在新建筑的三楼租用单独的公寓河边。该建筑有9套公寓，编号为301、302。 。 。 309。
奇数编号的公寓享有河景，而偶数编号的公寓则没有河景。吉尔只会租用河景的公寓，而杰克并不在乎。
他们最终租用的那对公寓有多少种不同的可能性？
2. [10分]
（a）如果n本身以外的所有n个因子的总和相等，则将n∈N称为完美数到例如，6是完美的，因为它的因子是1、2、3和6，并且1 + 2 + 3 = 6。
证明没有完美的质数。
（b）令m，n∈Z+，并假设m是n和n + 1的因数。证明m = 1。
解。
3. [10分] n≥2名杰出的霍格沃茨学生参加了斯内普教授的实验。
每个学生都会得到以下三种混合之一：药水A或药水B，或两者的混合物。
斯内普确保给哈利和赫敏各一个不同的调料。在多少
Snape可以通过各种不同的方式分配他的混合物吗？
4. [12分]证明对于所有奇数整数x和y我们有8 | X
2 − y
5. [8 pts]
(a) Give an example of three distinct (no two are the same), nonempty sets A, B, C such that
• there are elements that are common to A and B;
• every element of A that is also in B must also be in C;
• there are elements in A that are not in C.
(b) Let A be a finite set such that {∅} ∈ A and {∅} ⊆ A and |A| = 2. List all the subsets of
A. Justify your answer.
(c) Consider the sets A = {1, 2, 3}, B = {x
2
| x ∈ A}, and also C = {x+y | x ∈ B and y ∈ A}.
List the elements of A ∩ C. Show your work.
(d) Give examples of three sets A, B, C ⊆ {1, 2, 3, 4, 5, 6, 7} such that A and B are disjoint,
A \ C = {1, 3, 7}, B ∪ C = {2, 4, 5}, |A| = 5, and B \ C 6= ∅. Show your work.
6. [10 pts] Let A = {2, 3}, B = {3, 4}, C = {2, 3, 4}, and S = A × 2
B×2
C. Answer each of thefollowing questions. Explain your answers.
(a) (2, {(2, {2})}) ∈ S ?
(b) (2, {(3, {4})}) ∈ S ?
(c) (2, {({4}, 4)}) ∈ S ?