Let R and S be equivalence relations on a set X. Which of the following are necessarily equivalence relations?

(1)R ∩ S

(2)R \ S .

Please show me the proof. Thanks!

Use the elimination method to find a general solution for the given linear​ system, where differentiation is with respect to t.

x'=5x-6y+sin(t)

y'=3x-y-cos(t)

How can Singular Value Decomposition be used on image compression and its limitation?

Let A =

| 0.4 −0.3 |

| 0.4 1.2 |

Analyze the long-term behavior of Ak , i.e., lim k→+∞ Ak .

Consider the following function ?(?) = ?^ 4+ 2? ^3 + 8?^ 2+ 5? With the initial guesses of ?1 = −2, ?2 = −1, and ?3 = 1, find the minimum of the given function using parabolic interpolation. Perform five iterations, reporting Ɛa based on the location of the minimum (i.e. xopt) and not the actual minimum value. (Round the final answer to four decimal places.)

Show that {a + b√3 | a, b ∈ Q} is a field (a subfield of the field R), but {a + b√3 | a, b ∈ Z} is not a field.

An element a in a field F is called a primitive nth root of unity if n is the smallest positive integer such that aⁿ=1. For example, i is a primitive 4th root of unity in C, whereas -1 is not a primitive 4th root of unity (even though (-1)⁴=1).

(a) Find all primitive 4th roots of unity in F₅

(b) Find all primitive 3rd roots of unity in F₇

(c) Find all primitive 6th roots of unity in F₇

(d) Use Lagrange's Theorem to prove that if n does not divide p-1, then F_p contains no nth roots of unity. [In fact, the converse is true: If n divides p-1, then F_p contains a (primitive) pth root of unity. We will prove this later.]

Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R). Explain why {f(x) : f(0) = 1} is not.

Let S be a non-empty set (finite or otherwise) and Σ the group of permutations on S. Suppose ∼ is an equivalence relation on S. Prove (a) {ρ ∈ Σ : x ∼ ρ(x) (∀x ∈ S)} is a subgroup of Σ. (b) The elements ρ ∈ Σ for which, for every x and y in S, ρ(x) ∼ ρ(y) if and only if x ∼ y is a subgroup of Σ.

7. Suppose Bob has the public key (n, e) = (21733, 691). You are Eve, and you have intercepted the ciphertext C = 21012. On a whim, you decide to check whether C and n are relatively prime, and to your delight, you discover that they are not! Show how you can use this to recover the plaintext M.

Note: The chance that M (or equivalently C) is not relatively prime to the modulus n is1/p + 1 /q− 1/pq , and when p, q are both larger than 100 digits long, this probability is less than 10^−99! So although this is a vulnerability that one should be aware of, in practice it doesn’t cause issues very often.

A study was conducted to determine whether a new drug, nomasbesos, was effective in preventing the swollen spleen that often occurs when a person has a mononucleosis (mono) infection. Individuals with mono were randomly selected to either receive nomasbesos or a placebo treatment. Among the 315 who received the nomasbesos, 35 developed swollen spleens. Among the 275 individuals who received the placebo, 92 developed swollen spleens. Go out to 3 decimal points for all your calculations and then round to 2 decimal points for your final answer. The answers are below for your studying purposes.

Complete the 2x2 for the study. Be SURE to indicate precisely what your treatment and outcome are and complete all applicable cells.

1. What kind of study design (specifically) was used? How do you know?

2. What is the appropriate measure of association?

3. Calculate the value of the appropriate measure of association. Show all your work.

4. What does the calculated association mean in words?

Adirondack Savings Bank (ASB) has $1 million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are 5% for home loans, 13% for personal loans, and 8% for automobile loans. The bank’s planning committee has decided that at least 40% of the new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans.

Suppose a water tower in an earthquake acts as a mass-spring system. Assume that the container on top is full and the water does not move around. The container then acts as a mass and the support as a spring, where the induced vibrations are horizontal. Suppose that the container with water has a mass of m = 10, 000 kg. It takes a force of 1000 Newtons to displace the container 1 m. For simplicity assume no friction. When the earthquake hits, the water tower is at rest (not moving). Suppose the earthquake induces an external force F(t) = A cos(ωt)

Find the solution to the Initial Value Problem

An element a in a ring R is called nilpotent if there exists an n such that aⁿ = 0.

(a) Find a non-zero nilpotent element in M2(Z).

(b) Let R be a ring and assume a, b ∈ R have a^t = 0 and b^m = 0 for some positive integers t and m. Find an n so that (a + b)ⁿ = 0. (You just need to find any n that will work, not the smallest!)

(c) Show that the set of nilpotent elements in a commutative ring R forms a subring of R.

(d) Does this subring from the previous question contain a unity?

(e) Are the Gaussian integers from an earlier question an integral domain? Explain your answer.

Each time draw 6 different numbers from 1 ~ 45 randomly

Let M denote the times you have to draw such that all the number from 1~45 has been drawn
What's the expected number of M.

please explain thoroughly. Thanks.