If graph g has n vertices and k component and m edges, so m ≥ n-k. Prove it !
Thank you...

Determine the positive real root of ln(x^2)=0.8 by the following methods. (Note that you need to show the details of your derivations in MATLAB).

a) Graphically ( plot the function and copy your figure to word).

b) Using two iterations of the bisection method with initial guesses of xl=0.4 and xu=2 and populate the following table. What is the root after two iterations? Provide justification for the values you have obtained in your MATLAB code as comments.

i xl xu xr

1 0.4 2

2

c) Using two iterations of the false position method, with the same initial guesses as in b) and populate the table below. What is the root after two iterations? Provide justification for the values you have obtained in your MATLAB code as comments.

i xl xu xr

1 0.4 2

2   

d) Compute the actual root of the function (use a built in MATLAB function) and identify which method (bisection or false position) achieves a better estimate of the root after two iterations. Prove your answer by calculating the True Percentage Error ( Assume that your answer in part d, obtained using the MATLAB Built in Function is the true value ). ALL Calculations must be performed in MATLAB.

Lesson 7

• Explain the difference between population and a sample.
• Explain what an "unbiased sample" is.
• Explain the process for creating a frequency distribution.
• Explain how a histogram is drawn.

Prove the following T is linear in the following definitions
(a) T : R3 →R2 is defined by T(x,y,z) = (x−y,2z)
(b) T : R2 →R3 is defined by T(x,y) = (x−y,0,2x+y)
(c) T : P2(R) → P3(R) is defined by T(f(x)) = xf(x)+f(x)

TOPOLOGY

Let f : X → Y be a function.

Prove that f is one-to-one and onto if and only if f[A^c] = (f[A])^c for every subset A of X. (prove both directions)

Use the generating function to find the first five (5) Legendre polynomials and verify their your answer using Rodrigues's formula.

Let n be a positive integer. Prove that two numbers n2+3n+6 and n2+2n+7 cannot be prime at the same time.

Which of the following are groups? + And · denote the usual addition and multiplication of real numbers.

(G, +) with G = {2^ n | n ∈ Z},
(G, ·) with G = {2 ^n | n ∈ Z}.
Determine all subgroups of the following cyclic group
G = {e, a, a^2, a^3, a^4, a^5}.
Which of these subgroups is a normal divisor of G?

1) Solve the following problem graphically. Indicate (a) whether or not the problem is feasible, (b) whether or not the problem has an optimal solution, and (c) whether or not the problem is unbounded. If there is a unique optimal solution, specify the variable values for this solution. If there are 2 alternative optimal solutions, give the values for three different optimal solutions.
max 9x1 + 3x2
s.t. x2 ≤ 125
  − x1 + 2x2 ≤ 170
3x1 + x2 ≤ 300
− x1 + x2 ≥ 20
x1, x2 ≥ 0

2) PART A) Use the graphical approach to verify that the following problem is unbounded.
max 3x1 − x2
s.t. − 2x1 + x2 ≤ 0
  x1 + 2x2 ≥ 4
3x1 − 5x2 ≤ 10
  x1, x2 ≥ 0
PART B) Suppose you change the third constraint to “ax1 − 5x2 ≤ 10,” where a is nonnegative value. For what values of a does the problem (i) remain unbounded, (ii) have an optimal solution, and (iii) become infeasible?

Let ∆ABC be a triangle in R2 . Show that the centroid G is located at the

average position of the three vertices:

G = 1/3(A + B + C).

Express all solutions to the following equations as sets of integers: (a) 7x ≡ 12 mod 13 (b) 10x ≡ 4 mod 6 (c) 6x ≡ 8 mod 12

1. Seven thieves try to share a hoard of gold bars equally between themselves. Unfortunately, six bars are left over, and in the fight over them, one thief is killed. The remaining six thieves, still unable to share (all) the bars equally since two are left over, again fight, and another is killed. When the remaining five share (all) the bars, one bar is left over, and it is only after yet another thief is killed that an equal sharing is possible. What is the minimum number of bars which allows this to happen? [Hint: Be carefully to check that the conditions for the Chinese Remainder Theorem apply before using it.]

Differential Geometry

Open & Closed Sets, Continuity

(1) Prove (2,4) is open

(2) Prove [2,4) is not open

(3) Prove [2,4] is closed