5. Find a matrix A of rank 2 whose nullspace N(A) has dimension 3 and whose transposed nullspace N(AT) has dimension 2.

In how many ways can you distribute 10 different balls into 4 different boxes, so there's no box with exactly 3 balls? Use inclusion-exclusion

Monica bought a $2000 4K Ultra HD TV for 10% down, with the balance to be paid with interest at 13% compounded monthly in six equal monthly payments. Construct the full amortization schedule for the debt. Calculate the total interest paid.

Show that the transformation w = z ^1/ 2 maps usually maps vertical and horizontal lines onto portions of hyperbolas. Investigate the lines x = a, y = b for the cases

1) a > 0, b > 0,

2) a = b =0

NATURAL DEDUCTION

1. Construct a proof that reveals the validity of the following inference. Make sure that you symbolize it first.

Tom was involved in the fight at the party Friday and stole money from Monica’s purse later that same night. The parole officer had told me that if he was involved in the fight then he’d have to go back to the rehab clinic and the social worker told me that if he stole the money then control over his savings account is going to be turned over to a representative from the office of juvenile corrections. Well, it follows that Tom is going back into rehab and he’s going to lose control over his savings.

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Problem 4 [20 pts]: Playing Cards Let C represent a set of 52 playing cards with four suits (♥, ♦, ♣, ♠) each having 13 ranks (Ace,2,3,4,5,6,7,8,9,10,Jack,Queen,King). We define the following additional sets. F =Face cards (Jack, Queen, and King). R =Red cards. P =Ranks that are prime (2,3,5,7). J =One-eyed Jacks (Jack of Hearts, Jack of Spades).

1. Depict these sets as a Venn Diagram and show the cardinality of each distinct region. (The regions don’t have to be perfectly to scale - this can be hand-drawn.)

2. Using set notation, give an expression for the set of cards that are red or face cards or prime numbered or one-eyed Jacks and give its cardinality.

3. Give a set expression and cardinality for the set of cards that are not face cards or not prime-numbered cards.

4. Give a set expression and cardinality for the complement of the set of cards that are either red or prime-numbered but not one-eyed Jacks.

5. Give a set expression and cardinality for the set of cards that are either red non-prime cards or one-eyed Jacks, but not both.

Honeywell Bakery has been offering a new vegan cupcake and wants to know the demand in the future. For the past six weeks, the demand has been 60 cupcakes, 55 cupcakes, 75 cupcakes, 60 cupcakes, 80 cupcakes, and 75 cupcakes respectively. 1. a) Using a simple four-week moving average, calculate a forecast for next week. (4 points) b) If the forecast for week 6 was 85 cupcakes, then using single exponential smoothing with α=0.2, calculate a forecast for next week. (4 points) 2. a) Using simple linear regression, calculate the trend line for the historical data. X- Axis should represent the period (e.g. week 1 = 1, week 2 = 2, and so on), while the Υ axis is demand. (12 points) b) Calculate a forecast for next week. Is this forecast different from the ones calculated in 1a and 1b? If so, please explain the difference. (10 points)

Consider the following regression results of B for time series data. The dependent variable is the log of real consumption and the regressors are all lagged by one period. The coefficients are estimated by OLS method. *, **, and *** indicate that the coefficients are significant at the 10%, 5%, and 1% level, respectively. The F- test in the last row tests the joint hypothesis that all the coefficients except the constant are zero and their p-values are provided in the table. The empty space means that the variables are not included in the regression. (4) 3.1213*** 3.8876*** (1) Constant log(Real Income) 0.8683* Interest Rate Inflation Rate log(Stock Price) F-test 0.082 (2) (3) 3.3962*** 3.8115*** 0.6289*** 0.6341*** -0.2680** -0.2402** 0.5233 0.5230 -0.2504*** 0.6237 0.4775 0.064 0.095 0.044 (a) Why do we have to include lagg values of regressors in the equation? (b) Compare the results of equations (1) and (2) and make any comments. (c) Compare the results of equations (2) and (3) and make any comments. (d) Compare the results of equations (3) and (4) and make any comments.

Why does it not make sense for a function to have more than one output for the same input? Provide examples.
First , define inputs and outputs ( independent and dependent variables) . Then, define ordered pairs and relations and give examples. Finally, define a function and talk about what particular type of relation is a function.
Can you find a real world relation that is not a function?
For example: The age of a person is the input and his annual salary is the output. Two people who are the same age ( input) can have two different salaries ( output).
Can you find a real - world function that has more than one input leading to the same output?
For example : the height of a person is the input and his weight is the output.
Two people who have different heights ( input) can have the same weight ( output).

Assume G is an Abelian group of order 144 and G contains at least 5 elements of order 72. (a) Determine the possible structures of G (i.e., write it as a direct product of cyclic groups). If there are more than one structures, list them all. (b) For each isomorphic class of G, how many elements of order 12 does it have?

Find a closed formula for each of the following sequences. Show all work and explain your answers.

(a) {1, 6, 17, 34, 57, 86, 121, . . .}, where a₀ = 1.

(b) a_n = 5a_n−1 + 4, a₀ = 2

(c) a_n = 10a_n−1 − 21a_n−2, a₀ = 6, a₁ = 26.

Scenario

Implementing the sieve of Eratosthenes algorithm to find all prime numbers up to a given limit.

Aim

Develop code for implementing the sieve of Eratosthenes.

Steps for Completion

1. Implement the isPrime() method of the SieveOfEratosthenes class that should return true if the number is prime, and false otherwise.

2. Consider building the sieve in the class constructor.

CODE GIVEN

public class SieveOfEratosthenes {

public SieveOfEratosthenes(int maxValue) {

// Build the sieve here

}

public boolean isPrime(int value) {

// Write your code here

}

}

Assume a closed economy without Government. However, there exists a financial sector that creates an array of financial assets on which both households and firms invest.

Let ? denote the average earnings from these financial assets. The consumption expenditure of the households is influenced by their wage income and the financial income and is given by ? = ?(?, ?); ?? > 0, ?? > 0, where ??, ?? are partial derivatives of consumption with respect to income ? and financial earnings ? respectively. Similarly, the real investment expenditure of firms is given by ? = ?(?, ?); ?? > 0, ?? < 0, where ??, ?? are the partial derivatives of the real investment with respect to income and financial earnings. Using either the Keynesian cross model or the Multiplier analysis, answer the following questions.

(i) Derive the relationship between output ? and financial earnings ?, and examine the analytical conditions under which the relationship is positive ( ?? ?? > 0) and negative ( ?? ?? < 0).

(ii) Describe why the scenario where the expansion in output driven by the rise in financial earnings, i.e. when ?? ?? > 0, could make the economy unstable and vulnerable to crisis?