Assume A = R and the relation R ⊆ A × A such that for x, y, ∈ R, xRy if and only if sin2 x + cos2 y = 1. Prove that R is an equivalence relation and for any fixed x ∈ R, find the equivalence class x

prove or disppprove.

Suppose A & B are sets.

(1) A function f has an inverse iff f is a bijection.

(2) An injective function f:A->A is surjective.

(3) The composition of bijections is a bijection.

A tank originally contains 100 gal of fresh water. Then water containing

lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the tank at a rate of

6 gal/min,

with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min. (Round your answer to two decimal places.)

\int \frac{x^6+5x^4+4x-3}{\left(x^2+3x+5\right)\left(x^2+7x+12\right)}dx

Use laplace transforms to solve the following problem.

y' + 20y = 6sin 2x; y(0) = 6

Find all possible homomorphisms between Z and Z5.

Can you supply examples of logical operations you apply to your everyday experience?

Use matlab code for bisection method and regula falsi. Thank you!

1. An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below.

                                                   Medium         Cost Per Ad        # Reached          Exposure Quality

                                                        TV                   700                    8000                         35

                                                      Radio                 240                    3000                         50

                                                Newspaper             360                    5000                         25

If the number of TV ads cannot exceed the number of radio ads by more than 3, and if the advertising budget is $25,000. Develop the linear programming problem and solve the model that will maximize the number reached and achieve an exposure quality if at least 1000. (Hint: You need 3 decision variables to develop this LP problem.)

Vector operators

Let A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:,

rank(C^TAC) = rank(C),

Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )

Exercise 2.16: Consider the folowing data

HistoData

36     39     36    35    36    20    19

46     40     42    34    41    36    42

40     38     33    37    22    33    28

38     38     34    37    17    25    38

1. Find the number of clases needed to construct an histogram.
2. Find the class length.
3. Define nonoverlapping classes for a frequency distribution.
4. Tally the number of values in each class and develop a frequency distribution.
5. Draw a histogram for these data.
6. Develop a percent frequency distribution.

IMPORTANT NOTE: SHOW ALL THE CALCULATIONS - (YOU WILL NOT GET ANY CREDIT IF YOU DO NOT SHOW ALL THE CALCULATIONS)

• Write at least 2 references in APA style.

y"-2y'+y=cos2t

1. general solution of corresponding homongenous equation

2. particular solution

3.solution of initial value problem with initial conditions y(0)=y'(0)=0