Abstract Algebra

Let n ≥ 2. Show that S_n is generated by each of the following sets.

(a) S₁ = {(1, 2), (1, 2, 3), (1, 2, 3, 4), ..., (1, 2, 3,..., n)}

(b) S₂ = {(1, 2, 3, ..., n-1), (1, 2, 3, ..., n)}

Consider the integers from 1 to 10. Give the set of pairs (a, b) that corresponds to relation a ≡ b mod 1.

Prove Theorem 29.9 (Cantor). There are countably many algebraic numbers.In this project, you will prove this theorem.

Given following ODE's

1) x' = x / 1+t, with x(0) = 1 find x(2)

2) x' = t+x with x(0) = 1, find x(2)

3) x' = t-x, with x(1) =2 find x(3)

4) x' = t-x/t+x, with x(2) = 1, find x(4)

a) Solve each of the ODE's using Euler's method with h = 0.5, and calculate the relative error

i) x' = x/1+t: Approximation ____________________ Relative error: ________________

ii) x'= t+x; Approximation ____________________ Relative error: ________________

iii) x' = t-x; Approximation ____________________ Relative error: ________________

iv) x' = t-x/t+x; Approximation ____________________ Relative error: ________________

would you please sow me the steps?

Thank you.

Let W denote the set of English words. For u, v ∈ W, declare u ∼ v provided that u, v have the same length and u, v have the same first letter and u, v have the same last letter.

a) Prove that ∼ is an equivalence relation.

b) List all elements of the equivalence class [a]

c) List all elements of [ox]

d) List all elements of [are]

e) List all elements of [five]. Can you find more than 15?

f) Bonus. Find all three letter words x such that [x] has 5 elements.

A, B and C be sets.

(a) Suppose that A ⊆ B and B ⊆ C. Does this mean that A ⊆ C? Prove your answer. Hint: to prove that A ⊆ C you must prove the implication, “for all x, if x ∈ A then x ∈ C.”

(b) Suppose that A ∈ B and B ∈ C. Does this mean that A ∈ C? Give an example to prove that this does NOT always happen (and explain why your example works). You should be able to give an example where |A| = |B| = |C| = 2.

[Jordan Measure] Could you prove the following ?

Prove that the sets Q ∩ [0, 1] and [0, 1] \ Q are not Jordan measurable.

Determine whether the equation is exact. If it is exact, FIND THE SOLUTION. If not write NOT EXACT.

A) (y/x + 12x) + (lnx - 3)y' = 0 x>0

Solve the given initial value problem.

B)  (12x² + y − 1) − (14y − x)y' = 0,    y(1) = 0

y(x) =

Determine at least approximately where the solution is valid. (Enter your answer as an inequality for which the solution is valid when true.)

The solution is valid as long as: _________

C) Find the value of b for which the given equation is exact.

(ye^9xy + x) + bxe^9xyy' = 0

b =

Solve it using that value of b: ______

.

A set X is said to be closed under multiplication if for every x1,x2 ∈ X we have x1x2 ∈ X. Let A be the union of all bounded subsets X ⊆ R that are closed under multiplication. Does inf(A) exist? If it does, find it.

Let a be an element of a finite group G. The order of a is the least power k such that a^k = e.

Find the orders of following elements in S₅

a. (1 2 3 )

b. (1 3 2 4)

c. (2 3) (1 4)

d. (1 2) (3 5 4)

Nuclear Fuel Cycle:

Assuming that the price per SWU is $80 and the cost of conversion is $4/kgU, what is the
price of the U3O8 ($/lb U3O8) beyond which it will cost less to enrich the already mined, purified, and
converted (to UF6) tails that contain 0.2% U-235 rather than mine new uranium? Assume the product
will be 3% enriched in U-235 in either case and the new tails will be 0.1% (when the old tails are
enriched). Tails stored as UF6 cost nothing.

Let ?1=(1,0,1,0) ?2=(0,−1,1,−1) ?3=(1,1,1,1) be linearly independent vectors in ℝ4.

a. Apply the Gram-Schmidt algorithm to orthonormalise the vectors {?1,?2,?3} of vectors {?1,?2,?3}.

b. Find a vector ?4 such that {?1,?2,?3,?4} is an orthonormal basis for ℝ4 (where ℝ4  is the Euclidean space, that is, the inner product is the dot product).

Using Google sheets: On a spreadsheet show how to use the bisection method to solve the equation cos⁡(x)=x numerically to at least four decimal place accuracy