Find the Fourier coefficients of the following signal.

x(t) = 5 + 2sin(w0.t) + cos(2.w0.t) - 3sin(2.w0.t)

(1 point)

Suppose that

R1={(2,2), (2,3), (2,4), (3,2), (3,3), (3,4)},R1={(2,2), (2,3), (2,4), (3,2), (3,3), (3,4)},
R2={(1,1), (1,2), (2,1), (2,2), (3,3), (4,4)},R2={(1,1), (1,2), (2,1), (2,2), (3,3), (4,4)},
R3={(2,4), (4,2)}R3={(2,4), (4,2)} ,
R4={(1,2), (2,3), (3,4)}R4={(1,2), (2,3), (3,4)},
R5={(1,1), (2,2), (3,3), (4,4)},R5={(1,1), (2,2), (3,3), (4,4)},
R6={(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)},R6={(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)},

Determine which of these statements are correct.
Check ALL the correct answers below.

A. R3R3 is transitive
B. R2R2 is not transitive
C. R3R3 is reflexive
D. R5R5 is transitive
E. R5R5 is not reflexive
F. R4R4 is antisymmetric  
G. R6R6 is symmetric
H. R1R1 is not symmetric  
I. R1R1 is reflexive
J. R2R2 is reflexive  
K. R3R3 is symmetric
L. R4R4 is transitive
M. R4R4 is symmetric

Expand the function f(z) = 1 / (z + 1)(z − 3)

as a Laurent series about z = 0 in three regions: 1) |z| < 1, 2) 1 < |z| < 3 and 3) |z| > 3.

Decide, with justification, if the following set properties are true for any sets A, B, and C.

1. A∪(B∩C)⊆A.

2. A∪(B∩C)⊆B.

3. A ∩ B ⊆ A

4. B ⊆ A ∪ B
5. A ⊆ B ⇒ A ∪ B ⊆ B.

6. A ∪ B ⊆ B ⇒ A ⊆ B.

7. A⊆B⇒A∩B=A.

8. A ∩ B = A ⇒ A ⊆ B
9. A∩(B∪C)⊆A∪(B∩C)
10. A∪(B∩C)⊆A∩(B∪C)
11. A\(B∩C)=(A\B)∪(A\C)

How do we construct the Ito's Integral(also called the stochastic integral), and what are the properties of the integral?

Define the following predicates:

Real( x)   =   “x is a real number”.

Pos(x) =   “x is a positive real number.”

Neg(x) =   “x is a negative real number.”

Int(x) =   “x is an integer.”

Rewrite the following statements without using quantifiers. Determine which

statements are true or false and justify your answer as best you can.

a) Pos(0)

b) ∀x, Real(x) ∧ Neg(x) → Pos(−x)

c) ∀x, Int(x) → Real(x)

d) ∃x such that Real(x) ∧∼ Int(x)

1. Show the 4-quadrant position and then find the magnitude and angle of each of the following complex numbers: (12 Points)

(a) 3+j4 (b) -3+j4 (c) -3-j4 (d) 3-j4

A mass weighing 16 pounds stretches a spring

feet. The mass is initially released from rest from a point 5 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to

the instantaneous velocity. Find the equation of motion

x(t)

if the mass is driven by an external force equal to

f(t) = 20 cos(3t).

(Use

g = 32 ft/s²

for the acceleration due to gravity.)

Consider the permutation ρ = (1, 2, 6)(3, 4, 5) in S10. How many conjugates does ρ has in S10? Hence determine the order of Cρ(S10), the centralizer of ρ in S10. Now determine the order of Cρ(A10) by observing that there is an odd permutation in Cρ(S10). How many conjugates does ρ has as an element of A10?

A company surveyed 1000 people on their age and the number of jeans purchased annually. The results of the poll are shown in the table.

A person is selected at random. Compute the probability of the following.

(a) The person is over 25 and purchases 3 or more pairs of jeans annually.
  

(b) The person is in the age group 12-18 and purchases at most 2 pairs of jeans annually.
  

(c) The person is in the age group 19-25 or purchases no jeans annually.
  

(d) The person is older than 18 or purchases exactly 1 pair of jeans annually.
  

Suppose that ? and ? are subspaces of a vector space ? with ? = ? ⊕ ?. Suppose also that ??, … , ?? is a basis of ? amd ??, … , ?? is a basis of ?. Prove ??, … , ??, ??, … , ?? is a basis of V.

2. Let’s consider how the operations (∧) and (∨) relate to the quantifier (∃).

a) Give and explain an explicit subset S of real numbers and
predicates P(x) and Q(x) for which the statement below is false:43
∃x ∈ S, P(x) ∧ Q(x) ↔ [∃x ∈ S, P(x)] ∧ [∃x ∈ S, Q(x)].

Since {1, 2, . . . , 6} is the set of all possible outcomes of a throw with a regular die, the set of all possible outcomes of a throw with two dice is Throws := {1, 2, . . . , 6} × {1, 2, . . . , 6}. We define eleven subsets P2, P3, . . . , P12 of Throws as follows: Pk := {<m, n>: m + n = k} for k ∈ {2, 3, . . . , 12}. For example, P3 is the set of all outcomes for which the sum of the two numbers of dots thrown is 3.

(a) Show that the sets P2, P3, . . . , P12 form a partition of the set Throws.

(b) Let R be the equivalence relation on Throws that has P2, P3, . . . , P12 as its equivalence classes. Give a definition of R by means of a description.

(c) Give a complete system of representatives for the equivalence relation R.

a) Use truth tables to show that the following are valid arguments:

i. [p  (p → q)] → q

ii. [(p → q) ∧ (q → r)] → (p → r)

b) Use truth tables to show the logical equivalence of:

i. (p → q) ⇔ (￢p ∨ q )

ii. (￢p ∨ q) ∨ (￢p  q) ⇔ p

find the particular solution for:

y'' - 4y' - 12y = e^2x(-3x² + 4x + 5) (hint: try y= ue^2x then use: u = Ax² + Bx + C)