Questions
Let A/B = {w| wx ∈ A for some x ∈ B}. Show that if A...

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}

In: Advanced Math

Let A and B be sets. Then we denote the set of functions with domain A...

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.

In: Advanced Math

Write down every permutation in S3 as a product of 2-cycles in the most efficient way...

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.

In: Advanced Math

Find a particular solution to the equation 2y''+0.1y'+2y = cos(t) + 4cos(5t)

Find a particular solution to the equation

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

In: Advanced Math

Consider the letters ABCDE. a. How many strings of length 3 can be formed if we...

Consider the letters ABCDE.
a. How many strings of length 3 can be formed if we allow repetitions (if we allow a letter to be used more than once)?
b. How many strings of length 3 can be formed if we do not allow repetitions (if we allow each letter to be used at most once)?
c. How many strings of length 3 can be formed if the first letter is A and we allow repetitions?
d. How many strings of length 3 can be formed if the first letter is A we do not allow repetitions?

In: Advanced Math

Solve the following logic problems. Remember, everyone you meet is either a knight or a knave,...

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?

In: Advanced Math

1. Consider the following axiomatic system: A1: Each point is incident to exactly 2 lines. A2:...

1. Consider the following axiomatic system:
A1: Each point is incident to exactly 2 lines.
A2: For each pair of distinct lines, there is a point that is incident to both lines.
A3: There are exactly 5 lines.

(a) Pick one of the three axioms and prove that it is independent from the other axioms in the system. Be sure to justify your answer.

(b) Find the minimum number of points present in a model for this system. Be sure to justify your answer using the axioms given above.

In: Advanced Math

1. [9 marks] Consider the boundary-value problem, y′′ +2εy1/2 = 0, y(0) = 1,y(1) = 3/2....

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).

In: Advanced Math

1. Write two examples of media or types of communication in each section. Note that each...

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.

Mediated Communication

Not Mediated Communication

Mass Communication

Not mass communication

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

In: Advanced Math

do we have to pay pennsylvania state tax for early withdrawal of 1099-R.

do we have to pay pennsylvania state tax for early withdrawal of 1099-R.

In: Advanced Math

Consider the differential equation, L[y] = y'' + p(t)y' + q(t)y = 0, (1) whose coefficients...

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

y1(t) =
y2(t) =

In: Advanced Math

Find the solution of the given initial value problem y'' + 4y = t^2 + 3e^t...

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: Advanced Math

In control systems analysis, transfer functions are developed that mathematically relate the dynamics of a system’s...

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.]

In: Advanced Math

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

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 A2tA2.

n) Find rank(A2), rank(A2t), rank(TA2) and rank(A2tA2).

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!

In: Advanced Math

Prove the chain rule by defination.

Prove the chain rule by defination.

In: Advanced Math