Discrete math problem:

Prove that there are infinitely many primes of form 4n+3.

find the coefficient for the terms

(A) What is the coefficient for the term x 4y 3 in (x + y) 7 ?

(B) What is the coefficient for the term x 4y 3 in (x − y) 7 ?

(C) What is the coefficient for the term x 2y 3 z 2 in (x + y + z) 7 ?

1. Prove that the Cantor set contains no intervals.

2. Prove: If x is an element of the Cantor set, then there is a sequence X_n of elements from the Cantor set converging to x.

Number Theory

Exercise 1 Prove that the equation 3x^2 + 2 = y^2 has no solution (x, y) ∈ Z × Z. (Hint: consider the associated congruence modulo 3.)

Exercise 2 Prove that the equation 7x^3 + 2 = y^3 has no solution (x, y) ∈ Z × Z. (Hint: consider the associated congruence modulo 7.)

Let T : R2 → R3 be a linear transformation such that T( e⃗1 ) = (2,3,-5) and T( e⃗2 ) = (-1,0,1).

1. Determine the standard matrix of T.

2. Calculate T( ⃗u ), the image of ⃗u=(4,2) under T.

3. Suppose T(v⃗)=(3,2,2) for a certain v⃗ in R2 .Calculate the image of ⃗w=2⃗u−v⃗ .

4. Find a vector v⃗ inR2 that is mapped to ⃗0 in R3.

7. Show that the dual space H' of a Hilbert space H is a Hilbert space with inner product (', ')1 defined by
(f .. fV)1 = (z, v)= (v, z), where f.(x) = (x, z), etc.

I would like to know how to plot two graphs in matlab and find where they intercept.

Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous bijection. Prove that if (X, dX ) is compact, then f is a homeomorphism. (Hint: it might be convenient to use that a function is continuous if and only if the inverse image of every open set is open, if and only if the inverse image of every closed set is closed).

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V a subspace of R3?

Find the modulus of:

(a) (3 – j4) (-5 + j12)

(b) (2+?)/(4?+(1+?)^2)

Express (6 – ?8)^−3 in the standard form ? + ??. Find its conjugate.

If ? + ?? = (?+?)/ (?−?) ,where a, b and c are real, prove that ?^2 + ?^2 = 1 and ?/?= (2?)/ (?^2−1)

Please show all your steps so I can understand

Thank you

Present two cases of the Cardano’s formula

The legislature in a state has 59 seats. Apportion these seats to the five counties below using Jefferson's method. County Population Seats received

Adams 131,000

Grant 350,000

Colton 300,000

Davis 347,000

Hayes 236,000