Let F be a field.

1. (a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) = 2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple. That is, a^2 + b^2 = c^2. Use this fact to explain how to generate right triangles with integer side lengths.

2. (b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy − y^2 and c(x,y) = x^2 − xy + y2 in F[x,y] satisfy the equation a^2 − ab + b^2 = c^2. Use this fact to explain how to generate triangles with integer side lengths containing

   an angle of π /3

3. (c) Explain how to generate triangles with integer side lengths containing an angle of 2π / 3

For p, q ∈ S^1, the unit circle in the plane, let
d_a(p, q) = min{|angle(p) − angle(q)| , 2π − |angle(p) − angle(q)|}
where angle(z) ∈ [0, 2π) refers to the angle that z makes with the positive x-axis.
Use your geometric talent to prove that d_a is a metric on S^1.

what is the distinction between the terms sample and population. explain why sampling is necessary in some situations and census.(sampling the whole population) is necessary in some situations.

what are some factors that a manufacturer should consider when determining whether to test a sample or the entire population to ensure the quality of a product?

In Exercises 1–59 find a particular solution

y'''+3y''+4y'+12y=8cos2x - 16sin2x

Let B be a finite commutative group without an element of order 2. Show the mapping of b to b2 is an automorphism of B. However, if |B| = infinity, does it still need to be an automorphism?

For f: N x N -> N defined by f(m,n) = 2^m-1(2n-1)

a) Prove: f is 1-to-1

b) Prove: f is onto

c) Prove {1, 2} x N is countable

You recently acquired books on three different subjects in the following quantities: 6 history books, 5 music books, and 4 photography books.

(a) In how many ways can you arrange the books on a shelf?
(b) In how many ways can you arrange the books on a shelf so they can be grouped by subject?
(c) In how many ways can they choose 6 books with 2 books per subject?
(d) In how many ways can they choose 6 books with at least 4 history books?

Solve the discrete mathematics problem above using permutations/combinations. Show all work.

Find a particular solution y_p of the following EQUATIONS using the Method of Undetermined Coefficients. Primes denote the derivatives with respect to x.

y''-16y=cos h(4x)

y''+36y=12cos(6x)+18sin(6x)

y''+4y'+8y=325e^2tcos(5t)

y⁽⁵⁾+6y⁽⁴⁾-y=12

y⁽⁵⁾+2y⁽³⁾+2y''=8x²-2

SOLVE ALL ~ do ur besest (:

please proof and explain fundamental theorem of arithmetic for F[x] including results

(c) (¬p ∨ q) → (p ∧ q) and p

(d) (p → q) ∨ p and T

I was wondering if I could get help proving these expressions are logically equivalent by applying laws of logic.

Also these 2 last questions im having trouble with.
Rewrite the negation of each of the following logical expressions so that all negations
immediately precede predicates.
(a) ¬∀x(¬P(x) → Q(x))
(b) ¬∃x(P(x) → ¬Q(x))

Show that the set of rigid motions E(3) forms a group.

Define the probability density functions (PDF): Binomial, Uniform, and Normal distributions. Provide one example of each of them, and their graphs.