Let A/B = {w| wx ∈ A for some x ∈ B}. Show that if A is context free and B is regular, then A/B is context free.

Use the pumping lemma to show that the following languages are not context free. a. {0n1n0n1n| n ≥ 0}

Let A and B be sets. Then we denote the set of functions with domain A and codomain B as B^A. In other words, an element f∈B^A is a function f:A→B.

Prove: Let ?,?∈?^? (that is, ? and ? are real-valued functions with domain ?) and define a relation ≡ on ?^? by ?≡?⟺?(0)=?(0). (That is, ? and ? are equivalent if and only if they share the same value at ?=0) Then ≡ is an equivalence relation on  ?^?.

Prove: Suppose that ?:?→? f:A→B and ?,?⊆? Then ?(?∩?)⊆?(?)∩?(?)

Prove: Suppose that ?:?→? f:A→B and E,F⊆B. Then ?^(−1)(?)∪?(−1)(?)⊆?(?∪?)

Prove: Suppose that {Ai} is a partititon of B. Then the relation x∼y for defined x,y∈A by x∼y⟺∃Ai(x,y∈Ai) is an equivalence relation.

Write down every permutation in S3 as a product of 2-cycles in the most efficient way you can find (i.e., use the fewest possible transpositions). Now, write every permutation in S3 as a product of adjacent 2-cycles, but don’t worry about whether your decomposition are efficient. Any observations about the number of transpositions you used in each case? Think about even versus odd.

Find a particular solution to the equation

2y''+0.1y'+2y = cos(t) + 4cos(5t)

Solve the following logic problems. Remember, everyone you meet is either a knight or a knave, knights make true statements, and knaves make false statements. Give your reasoning for each problem.

a) You meet two residents, Alex and Bill. They say the following: Alex: I’m a knight. Bill: Alex is a knight, but I’m a knave. Is Alex a knight or a knave? Is Bill a knight or a knave?

b) You meet Clara and Davis, who are all like: Clara: One of us is a knight and the other is a knave. Davis: Clara is a knave. Is Clara a knight or a knave? Is Davis a knight or a knave?

c) You meet Edith and Frank, though only Edith speaks. Edith: Both Frank and I are knaves. Is Edith a knight or a knave? Is Frank a knight or a knave? (Note: Frank’s silence gives no indication of his type, but you can figure out from Edith’s statement.)

d) You meet Gina, Herbert, and Ichabod. Gina: Ichabod is a knave, if and only if I’m a knight. Herbert: Ichabod is a knight, if and only if I’m a knave. Ichabod: I like pudding. Does Ichabod like pudding?

1. [9 marks] Consider the boundary-value problem,
y′′ +2εy1/2 = 0, y(0) = 1,y(1) = 3/2.
Letting y = y0 + εy1 + ε2y2 + . . . , find y0 and y1 and hence y with error O(ε2).

1. Write two examples of media or types of communication in each section. Note that each section is described by the column header AND the row title. Leave a section blank if, by definition, no example is possible.

2. What specifically distinguishes mass communication from earlier means of communication?

Consider the differential equation,

L[y] = y'' + p(t)y' + q(t)y = 0, (1)

whose coefficients p and q are continuous on some open interval I. Choose some point t₀ in I. Let y₁ be the solution of equation (1) that also satisfies the initial conditions

y(t₀) = 1,

y'(t₀) = 0,

and let y₂ be the solution of equation (1) that satisfies the initial conditions

y(t₀) = 0,

y'(t₀) = 1.

Then y₁ and y₂ form a fundamental set of solutions of equation (1).

Find the fundamental set of solutions specified by the theorem above for the given differential equation and initial point.

y'' + 8y' − 9y = 0,

t₀ = 0

Find the solution of the given initial value problem y'' + 4y = t^2 + 3e^t + e^2t cost, y(0) = 0, y'(0) = 2,

using method of undetermined coefficients

In control systems analysis, transfer functions are developed that mathematically relate the dynamics of a system’s input to its output. A transfer function for a robotic positioning system is given by ?(?) = ?(?) ?(?) = ?3+12.5?2+50.5?+66 ?4+19?3+122?2+296?+192 . Where, ?(?) = system gain, ?(?) = system output, ?(?) = system input, and ? = Laplace transform complex frequency. Now, use Bairstow’s method to determine the roots of the numerator and denominator and factor these into the form ?(?) = (?+?1)(?+?2)(?+?3) (?+?1)(?+?2)(?+?3)(?+?4) . Where, ?? and ?? = the roots of the numerator and denominator, respectively. [Hints: To perform the evaluation of complex roots, Bairstow’s method divides the polynomial by a quadratic factor ?2 − ?? − ?. Use initial guesses of ? = ? = −1, for determining the roots of both numerator and denominator, and perform up to four iteration.]

Matrix A2= [1 2 3; 4 5 6; 7 8 9; 3 2 4; 6 5 4; 9 8 7]

Note: TA2 is defined to be a linear transformation that maps any vector x to A2* x. That is TA2 = A2*x. Also the range of the Linear transformation represented by A2 is the same as the column space of A2.

l) Find a basis for the null(TA2).

m) Find nullity of A2, TA2 and A2^tA2.

n) Find rank(A2), rank(A2^t), rank(TA2) and rank(A2^tA2).

I don't need any explanation, a concise answer is good enough. Please only help if you can give a full and correct answer, thank you!