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###### 日期：2020-09-05 08:47

CIT 592 Spring 2020 Homework 1
1. [10 pts] Jack and Jill want to rent separate apartments on the third floor of a new building
by the river. The building has nine apartments available, numbered 301, 302,. . . , 309. The
odd-numbered apartments have a river view, and the even-numbered apartments do not. Jill
will only rent an apartment with a river view, and Jack does not care about the view.
How many distinct possibilities exist for the pair of apartments they end up renting?
Solution.
2. [10 pts]
(a) A number n ∈ N is called perfect if the sum of all of n’s factors other than n itself is equal
to n. For example, 6 is perfect because its factors are 1, 2, 3, and 6, and 1 + 2 + 3 = 6.
Prove that there are no perfect prime numbers.
(b) Let m, n ∈ Z
+, and suppose that m is a factor of both n and n + 1. Prove that m = 1.
Solution.
Homework 1 CIT 592 2
3. [10 pts] n ≥ 2 distinguishable Hogwarts students participate in Professor Snape’s experiment.
Each student is given one of three concoctions: potion A, or potion B, or a mixture of the two.
Snape makes sure to give a different concoction to each of Harry and Hermione. In how many
distinct ways could Snape have distributed his concoctions?
Solution.
4. [12 pts] Prove that for all odd integers x and y we have 8 | x
2 ? y
2
.
Solution.
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
(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.
Solution.
Homework 1 CIT 592 3
6. [10 pts] Let A = {2, 3}, B = {3, 4}, C = {2, 3, 4}, and S = A × 2
B×2
C
(a) (2, {(2, {2})}) ∈ S ?
(b) (2, {(3, {4})}) ∈ S ?
(c) (2, {({4}, 4)}) ∈ S ?
Solution.