Prove

p → (q ∨ r), q → s, r → s ⊢ p → s

2. In each of the following problem sketch the graph of f(y) versus y, determine the equilibrium solutions, and classify each one as asymptotically stable, asymptotically unstable, or semi-stable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. Here y0 = y(0).

(a) dy/dt = e ^y − 1, −∞ < y0 < ∞

(b) dy/dt = (e^-y) − 1, −∞ < y0 < ∞

(c) dy/dt = (y^2)* (1 − y)^ 2 , −∞ < y0 < ∞

(d) dy/dt = y(1 − y ^2 ), −∞ < y0 < ∞

Let G be a connected graph which contains two spanning trees T1 and T2 that do not share any edges (note that G may contain edges that are in neither trees). Prove that G does not have a bridge.

I have a hint for the answer: Show that each edge is in a cycle (2 cases). But I can't figure out the 2 cases. Some help please!

Exercise 1. Establish the following logical equivalencies where the domain of P(x) is non-empty and Adoes not depend upon x:
i) ∀x(A → P(x)) ≡ A → ∀xP(x).
ii) ∀x(P(x) → A) ≡ ∃xP(x) → A.

A company manufactures two different types of gloves: a regular model and a catcher’s mitt.

The table below gives the basic information. Write the objective function, along with the constraints. All times are in hours.

Show linear dependence or independence. Show all steps algebraically.

a. let v1= < x1, x2, ... , xn > and v2 = < y1, y2, ... , yn > be vectors in R^n with v1 not equal to 0. Prove that v1 and v2 are linearly dependent if and only if v1 is a non-zero multiple of v2.

b. Suppose v1, v2, and v3, are linearly independent vectors in a vector space V. Show that w1, w2, w3, are linearly independent where w1 = v1 + v2 + v3

w2 = v1 - v2 - v3

w3 = 2v1 + v2 - v3

Hint: Assume that c1w1 + c2w2 + c3w3 = 0 and show that c1 = c2 = c3 = 0 by replacing w1, w2, w3 in the above equation with their expression in terms of v1, v2, v3, and use the fact that v1, v2 and v3 are linearly independent.

c. Suppose S = { v1, v2, ... , vn } is linearly independent. Prove that any non - empty subset of S is also linearly independent. Hint: Assume a subset w1, w2, ... , wk of S is linearly dependent. Show that this implies S is linearly dependent which is a contradiction.

for a and c yn and vn , the n are subscripts, and the numbers after the variables are subscripts, i wasnt sure how to type it here so v1 is v subscript 1. Thank you! Sorry for the confusion!

Let E be the set of all positive integers. Define m to be an "even prime" if m is even but not factorable into two even numbers. Prove that some elements of E are not uniquely representable as products of "even primes."

Please be as detailed as possible!

Calculate the first three terms in the power series solutions of the following differential equations taken about x=0.

x^2y''+sinx y'-cosx y=0

4 -letter words'' are formed using the letters A, B, C, D, E, F, G. How many such words are possible for each of the following conditions?

(a) No condition is imposed.
Your answer is :
(b) No letter can be repeated in a word.
Your answer is :
(c) Each word must begin with the letter A.
Your answer is :
(d) The letter C must be at the end.
Your answer is :
(e) The second letter must be a vowel.
Your answer is :

A natural cubic spline S is defined by S(x) = { S0(x) = a0 + b0(x − 1) + d0(x − 1)3 , if 1 ≤ x ≤ 2, S1(x) = a1 + b1(x − 2) − 3 4 (x − 2)2 + d1(x − 2)3 , if 2 ≤ x ≤ 3. Use S to interpolate data f(1) = 1, f(2) = 1, f(3) = 0, find a0, b0, d0, a1, b1, and d1.

How many times is line (5) executed in the following pseudocode? Enter your answer in the box below.

NOTE: Please read the pseudocode very carefully.

(1) n=14n=14

(2) m=16m=16

(3) for i=1i=1 to n+2n+2

(4) ---- for j=1j=1 to mm

(5) -------- print (i,j)

True/False Question: If sppan{u,v}=W where u not equal to v. Then dim(W)=2.

Answer: False

Reasoning let u=1, v=2 then span(1,2}=R but dim(R)=1 not 2.

I know the answer is false. please tell me whether my reasoning is correct.

I live in Rapid City and commute to Spearfish. I leave Rapid City by 6 minutes after 8 and get onto I-90 at Exit 52. I must exit I-90 at Exit 12 by no later than 8:50 so that I am not late for class. Normally, I can drive the speed limit and make it on time.

Over the past few years there has been a lot of construction on Interstate 90. For example, last year 4 miles of road was under construction between Mile Marker 26 and 30 (Sturgis). The posted speed limit in the construction zone (single lane -- do not pass) is 55 mph, but I always end up behind someone that is going only 49mph. When I am not in the Construction Zone, what is the minimum speed that I have to drive, so that I can be at Exit 12 by 8:50?

Please solve all parts

Consider the following IVP

y′ = 2t − y + 1, y(0) = 1.

Find an approximation of y(1) using

(a) the Euler’s method with N = 4,

(b) the Euler’s method with N = 8,

(c) the improved Euler’s method with N = 4,

(d) the improved Euler’s method with N = 8.

Solve the IVP, find y(1), and compare the accuracy |y(1)−yN | of the approximations.