1. Suppose ?:? → ? and {??}?∈? is an indexed collection of subsets of set ?. Prove ?(⋂ ?? ?∈? ) ⊂ ⋂ ?(??) ?∈? with equality if ? is one-to-one.

2. Compute:

a. ⋂ ∞ ?=1 [?,∞)

b. ⋃ ∞ ?=1 [0,2 − 1 /?]

c. lim sup ?→∞ (−1 + (−1)^? /?,1 +(−1)^? /?)
d. lim inf ?→∞(−1 +(−1)^?/ ?,1 +(−1)^? /?)

Determine the truth value of the following statements if the universe of discourse of each variable is the set of real numbers.

  1. ∃x(x2=−1)∃x(x2=−1)

  2. ∃x∀y≠0(xy=1)∃x∀y≠0(xy=1)

  3. ∀x∃y(x2=y)∀x∃y(x2=y)

  4. ∃x∃y(x+y≠y+x)∃x∃y(x+y≠y+x)

  5. ∃x∀y(xy=0)∃x∀y(xy=0)

  6. ∀x∃y(x=y2)∀x∃y(x=y2)

  7. ∀x∀y∃z(z=x+y2)∀x∀y∃z(z=x+y2)

  8. ∀x≠0∃y(xy=1)∀x≠0∃y(xy=1)

  9. ∃x(x2=2)∃x(x2=2)

  10. ∀x∃y(x+y=1)∀x∃y(x+y=1)

  11. ∃x∃y((x+2y=2)∧(2x+4y=5))∃x∃y((x+2y=2)∧(2x+4y=5))

  12. ∀x∃y((x+y=2)∧(2x−y=1))

Find the roots of the following equation in [−π, π] 2x 2 − 4 cos(5x) − 4x sin x + 1 = 0 by using the Newton’s method with accuracy 10^(−5) .

how do I solve this using a computer

Find the solution of the initial value problem:

y'' + 4y' + 20y = -3sin(2x), y(0) = y'(0) = 0

Question 1

A) Show that the functions y1(t) = 1 + t 2 ; y2(t) = 1 − t 2 are linearly independent directly from the definition of linear independence.

B)Find three functions y1(t), y2(t), y3(t) such that any two of them are linearly independent but three of them are not linearly independent.

Write a Matlab function for a matrix that takes in a matrix in echelon form and will return the row canonical form. The function cannot use rref, or any other matlab built in functions.

Compute the center Z of SL2(Z/p)

1. As dry air moves upward, it expands and, in so doing, cools at a rate of about 1oC for each 100-meter rise, up to about 12 km. If the ground temperature is 10oC on the ground, what range of temperatures can be expected if an airplane takes off and reaches a maximum height of 4 km? (1 km = 1000 meters

2. What amount of a 15% HCL acid solution must be mixed with a 20% HCL acid solution to obtain 50 milliliters of 18% solution?

3. A telephone company offers two long-distance plans:Plan A: $25 per month and $.05 per minutePlan B: $5 per month and $.12 per minuteFor how many minutes of long-distance calls would Plan B be financially advantageous?

Why is Gauss Elimination faster than solving a system of linear equations by using the inverse of a Matrix? (I know it has something to do with there being less operation with Gauss elim.) Can you show an example with a 2x2 and 3x3 matrix?

Let A and B be groups, and consider the product group G=A x B.

(a) Prove that N={(e_a,b) E A x B| b E B} is a subgroup.

(b) Prove that N is isomorphic to B

(c) Prove that N is a normal subgroup of G

(d) Prove that G|N is isomorphic to A

. Solve the Initial value problem by using Laplace transforms: ? ′′ + 3? ′ + 2? = 6? −? , ?(0) = 2 ? ′ (0) = 8

Find a basis and the dimension of W. Show algebraically how you found your answer.

a. W = {(x1, x2, x3, x4) ∈ R^4 | x2 = x3 and x1 + x4 = 0}

b. W = {( A ∈ M 3x3 (R) | A is an upper triangular matrix}

c. W = { f ∈ P3 (R) | f(0) = 0.

Show that if Y is a subspace of X, and A is a subset of Y, then the subspace topology on A as a subspace of Y is the same as the subspace topology on A as a subspace of X.