Question

Is the set A={k sin(2k-1) | k=-3,-2,-1,0,1,2,3,4,5,…} countable? Justify your answer.

Answer #1

1. Prove that {2k+1: k ∈ Z}={2k+3 : k ∈ Z}
2. Prove/disprove: if p and q are prime numbers and p < q,
then 2p + q^2 is odd (Hint: all prime numbers greater than 2 are
odd)

Show that the equation
x+sin(x/3)−8=0
has exactly one real root. Justify your answer.

We often reason with trigonometric identities such as sin^2(x) +
cos^2(x) = 1. Given a set of such identities S and an identity to
prove using the set S, what would be the most efficient data
structure and algorithm for this purpose. Justify your answer.
Explain the representation of the identities with an example.

c.) Determine whether the seriesX∞ k=1 k(k^4 + 2k)/(3k 2 − 7k^5)
is convergent or divergent. If it is convergent, find the sum.
d.) Determine whether the series X∞ n=1 n^2/(n^3 + 1) is
convergent or divergent.

Define the set E to be the set of even integers; that is,
E={x∈Z:x=2k, where k∈Z}. Define the set F to be the set of integers
that can be expressed as the sum of two odd numbers; that is,
F={y∈Z:y=a+b, where a=2k1+1 and
b=2k2+1}.Please prove E=F.

tan plane at (1,1)
v(j,k)= ln(10j^2 2k^2 + 1)

5.
(a) Find the values of 2k mod 9 where k = 1, 2, 3, 4, 5,
6, 7, 8, 9.
(b) Find the remainder of 2271 when this number is
divided by 9.
(c) Find the remainder of 2271 when this number is
divided by 5.
PLEASE GIVE RIGHT ANSWERS AND GOOD DETAILS ON HOW TO GET THE
ANSWERS
THE ANSWERS I GOT YESTERDAY FROM YOUR GUYS WERE WRONG!!!

If p = 2k − 1 is prime, show that k is an odd integer or k =
2.
Hint: Use the difference of squares 22m − 1 = (2m − 1)(2m +
1).

Prove that the set of real numbers of the form e^n,n=
0,=+-1,+-2,... is countable.

Let m = 2k + 1 be an odd integer. Prove that k + 1 is the
multiplicative inverse of 2, mod m.

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