Find the minimal generating set of G

-- G = Z2 × Z3

-- G = Z2 × Z2

-- G = Z × Z

(a) Can the sum of 12345 odd pos-
itive integers be 12345678 ? Prove or disprove.
(b) Prove that for every possible re-arrangement (b1, b2, . . . , b999) of the
numbers (1, 2, . . . , 999), the product
(b1 − 1)(b2 − 2)(b3 − 3) · . . . · (b999 − 999) is even.

Are the groups D2 × A3 and Z/6Z × S2 isomorphic?

use truth tables to determine whether or not the following arguments are valid:

a) if jones is convicted then he will go to prison. Jones will be convicted only if Smith testifies against him. Therefore , Jones won't go to prison unless smith testifies against him.

b) either the Democrats or the Republicans will have a majority in the Senate. but not both. Having a Democratic majority is a necessary condition for the bill to pass. Therefore, if the republicans have a majority in the senate there the bill won't pass.

a. Solve 7x + 5 ≡ 3 (mod 19).

b. State and prove the Chinese Remainder Theorem

c. State and prove Euler’s Theorem.

d. What are the last three digits of 9^1203?

e. Identify all of the primitive roots of 19.

f. Explain what a Feistel system is and explain how to decrypt something encoded with a Feistel system. Prove your result.

(a) a sequence {a_n} that is not monotone (nor eventually monotone) but diverges to ∞
(b) a divergent sequence {a_n} such that {a_n/33} converges
(c) two divergent sequences {a_n} and {b_n} such that {a_n + b_n} converges to 17
(d) two convergent sequences {a_n} and {b_n} such that {a_n/b_n} diverges
(e) a sequence with no convergent subsequence
(f) a Cauchy sequence with an unbounded subsequence

Find the solution of

y″+8y′=1024sin(8t)+640cos(8t)

y(0)=4 and y'(0)=3

y=?

Identify the flaws in the reasoning of the researchers in the following two scenarios, and discuss these flaws as completely as you can given the information provided in the question.

(a) Five hundred students at a randomly selected big-city elementary school are given a 25-question survey that they are to return the next day after their parents have filled it out. One of the questions asked of parents is, “Does your work schedule make it difficult to spend after-school time with your kids?” Among the parents who replied, 85% said “no”. Based on these results, the researchers concluded that the great majority of parents have no difficulty spending time with their kids after school.

(b) A survey is conducted on a random sample of 1,000 women who recently gave birth, asking them whether they smoked during the pregnancy. A follow-up survey 3 years later asked if their children have had any respiratory problems. Of the original 1,000 women who gave birth, 567 were reached at the same address and responded to the follow-up survey. The researcher reported that these 567 women are representative of all mothers with children under 3 years old.

Recall the group Hom(G, A) (especially the group Hom(G, C ∗ ) whose elements are called characters of G) and the group µn of n-th roots of unity.

(i) Let n be a positive integer, prove that Hom(Cn, C ∗ ) ∼= µn. Hints: let g be a generator of Cn. For every homomorphism α : Cn → C ∗ , prove that α(g) ∈ µn (i.e. α(g) is an n-th root of unity). Hence we have the map Hom(Cn, C ∗ ) → µn given by α 7→ α(g). Prove that this map is a group homomorphism and it is bijective.

(ii) Let G1 and G2 be groups, prove that Hom(G1 × G2, C ∗ ) ∼= Hom(G1, C ∗ ) × Hom(G2, C ∗ ).

Q1: A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of 56 women over the age of 50 used the new cream for 6 months. Of those 56 women, 30 of them reported skin improvement(as judged by a dermatologist). Is this evidence that the cream will improve the skin of more than 40% of women over the age of 50? Test using α=0.01.

(a) Test statistic: z=

(b) Critical Value: z∗=

(c) The final conclusion is

A. We can reject the null hypothesis that p=0.4 and accept that p>0.4. That is, the cream can improve the skin of more than 40% of women over 50.
B. There is not sufficient evidence to reject the null hypothesis that p=0.4. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than 40% of women over 50.

Q2: A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of n=1300 registered voters and found that 670 would vote for the Republican candidate. Let pp represent the proportion of registered voters in the state who would vote for the Republican candidate.
We test

H0:p=.50
Ha:p>.50

(a) What is the z-statistic for this test?  

(b) What is the P-value of the test?  

Find the inverse Laplace transform of

a. F(s) =  se^-6s/(s-2)(s²+2s+1)

b. F(s) = e^-12s/s³(s²+4)

Problem 1

1.1 If A is an n x n matrix, prove that if A has n linearly independent eigenvalues, then A^T is diagonalizable.

1.2 Diagonalize the matrix below with eigenvalues equal to -1 and 5.

1.3 Assume that A is 4 x 4 and has three different eigenvalues, if one of the eigenspaces is dimension 1 while the other is dimension 2, can A be undiagonalizable? Explain.

Answer for all 3 questions required.

Let X ∈ L(U, V ) and Y ∈ L(V, W). You may assume that V is finite-dimensional.

1)Prove that dim(range Y) ≤ min(dim V, dim W). Explain the corresponding result for matrices in terms of rank

2) If dim(range Y) = dim V, what can you conclude of Y? Give some explanation

3) If dim(range Y) = dim W, what can you conclude of Y? Give some explanation

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f is non-decreasing on [a,b] if and only if f′(x) ≥ 0 for all x ∈ (a,b), while if f is non-increasing on [a,b] if and only if f′(x) ≤ 0 for all x ∈ (a, b).

can you please prove this theorem? thank you!