Question 1

a) Prove that if u and v are distinct vertices of a graph G, there exists a walk from u to v if and only if there exists a path (a walk with distinct vertices) from u to v.

b) Prove that a graph is bipartite if and only if it contains no cycles of odd length.

Please write legibly with step by step details. Many thanks!

Consider the following. W(r, h) = 2.8r2(1.08h) + 59r(0.3h2 − 3.3h + 72) when r = 13, h = 63, and Δr = −1.5 (a) Calculate the output associated with the given input values. (Round your answer to three decimal places.) W(13, 63) = (b) Approximate the change needed in one input variable to compensate for the given change in the other input variable. (Round your answer to three decimal places.) Δh =

Write and submit at a Matlab function that sums up a specified number of odd-power terms from the Maclaurin series of sin(x)  using following function format

function s = etaylor(x, n)

% Sum of first n non-vanishing terms of Maclaurin series of sin(x)
% For example, etaylor(0.5, 1) should be 0.5, and etaylor(0.5, 2) should be 23/48
  

Suppose we are given the unit square S in the plane with corners (0, 0), (1, 0), (1, 1) and (0, 1). Let T : R ² → R ² be a linear transformation represented by the matrix A. If T is not onto, will the image of S under the map T be a parallelogram, a line segment, or a point? Be sure to justify your answer.

Consider any two finite sets A and B. Prove that |A×B|=|A||B|

Prove that if A and B are enumerable, so is A x B.

3. Write down the multiplication tables for the direct products

   (i) Z4×Z2;
   (ii) G5×G3;
   (iii) Z2 ×Z2 ×Z2;
   (iv) G12×G4. Which of the above groups are isomorphic to each other?

1. You deposit $4000 in an account earning 5% interest compounded monthly. How much will you have in the account in 5 years?

2. Find the time required for an investment of 5000 dollars to grow to 6500 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
Round your answer to two decimal places
Your answer is t=____ years.

3. You deposit $2000 in an account earning 5% interest compounded monthly. How much will you have in the account in 15 years?

Determine the roots of the following simultaneous nonlinear equations using multiple-equation Newton Raphson method. Carry out two iterations with initial guesses of

x₁⁽⁰⁾ =0.6 and x₂⁽⁰⁾ =1.2. Calculate the approximate relative error ε_a in each iteration by using maximum magnitude norm (║x║∞).

x₁ + 1 - x₂² = 0

x₁² + x₂² – 5 = 0

Suppose that a subset S of an ordered field F is not bounded above in F. Let T be a subset of F satisfying the property that, for each x ∈ S, there exists y ∈ T such that x ≤ y. Prove that T is not bounded above in F.

A company must meet (on time) the following demands quarter 1|30 units; quarter 2|20 units; quarter 3|40 units. Each quarter, up to 27 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlimited number of units can be produced with overtime labor, at a cost of $60 per unit. Of all units produced, 25% are unsuitable and cannot be used to meet demand. Also, at the end of each quarter, 10% of all units on hand spoil and cannot be used to meet any future demands. After each quarter's demand is satisfied and spoilage is accounted for, a cost of $15 per unit is assessed against the quarter's ending inventory. Formulate an LP that can be used to minimize the total cost of meeting the next three quarters' demands. Assume that 20 usable units are available at the beginning of quarter 1. (Hint: Define inventory variables to keep track of the number of usable units.)

Find Eigenvalues and eigenvectors

6 -2 2

2    5 0

-2    0 7

The Swiss mathematician Leonard Euler pronounced, Oiler, who lived from 1707 to 1783. Do some research and write two paragraphs on something you found out about this mathematician. It could be about his mathematical career, his life, etc.

A projective plane is a plane (S,L ) satisfying the following four axioms. P1. For any two distinct points P and Q there is one and only one line containing P and Q. P2. For any two distinct lines l and m there exists one and only one point P belonging to l∩m. P3. There exist three noncollinear points. P4. Every line contains at least three points.

Let π be a projective plane. Using P1 − P4, show that π contains two distinct lines l and m and a point P such that P does not lie on l or m.