You have just started a new job that offers a retirement savings account. You have two options: You can invest 5% of your monthly wages at 2% OR You can invest 4% of your monthly wages at 4%. Both are compounded monthly. b. Assume that you will always make $45,000 annually, how much will you have saved with the better plan after 15 years? c.Assume that you will always make $45,000 annually, how much will you have saved with the better plan after 25 years?

Exercise 4.8. For each of the following, state whether it is true or false. If

true, prove. If false, provide a counterexample.
(i) LetX beasetfromRn. ThesetX isclosedifandonlyifX isconvex.

(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex then Xand Y are both closed and convex sets.

(iii) LetX beanopensetandY ⊆X. IfY ≠∅,thenY isaconvexset.

(iv) SupposeX isanopensetandY isaconvexset. IfX∩Y ⊂X then

X∪Y isopen.

Prove If C is a binary self-dual  code, show that every codeword has even weight. Furthermore, prove if each row of the generator matrix of C has weight

divisible by 4, then so does every codeword.

B. Five bowls are labeled 1,2,3,4,5. Bowl i contains i white and 5 − i black ping pong balls, for i = 1,2,3,4,5. A bowl is randomly selected, and 2 ping pong balls are selected from that bowl at random without replacement. Both selected balls were white. What is the probability they were selected from bowl 1? 2? 3? 4? 5?

B. Five bowls are labeled 1,2,3,4,5. Bowl i contains i white and 5 − i black ping pong balls, for i = 1,2,3,4,5. A bowl is randomly selected, and 2 ping pong balls are selected from that bowl at random without replacement. Both selected balls were white. What is the probability they were selected from bowl 1? 2? 3? 4? 5?

Prove that the minimum distance of a linear code is the minimum weight of any nonzero codeword.

Match the following:

Let X = {1,2,3,4}, Classify the relations of X on X
___ {(1,4),(1,2)}
___{ (1,4),(4,1),(2,3) }
___{ (1,4),(4,4),(2,3),(3,3)}
___{ (1,1),(4,4),(2,2),(3,3) }

a. Is a function
b. Is a relation
c. Is transitive
d. Is a relation of equivalence
e. Is not a function

1. Show that 11,111,111 and 3,333,333 are relatively prime using the Extended Euclidean Algorithm.

2. Use the EEA to find the GCD of 6,327 and 10,101.

3. Find the additive inverse of 3,333,333 modulo 11,111,111. Verify.

4. Find the multiplicative inverse of 3,333,333 modulo 11,111,111.Verify.

5. What is the orbit of 3 in the group Z_7 under multiplication modulo 7? Is 3 a generator?

Find either a linear or an exponential function that models the data in the table.

​f(x)=

​(Use integers or fractions for any numbers in the​ expression.)

Note: Please show how to solve using TI-84 Plus calculator

Determine if the column vectors [2,1,3,4]' [1,1,1,1]' and [5,3,7,9]' are linearly independent

(DRUG DOSAGE PROBLEM) A drug company wants to know how to calculate a suitable dose and time between doses to maintain a safe but effective concentration of a drug in the blood. To be simple for the users of the drug, only a fixed dose at regular time intervals is considered possible. Suggest a way to model the problem, determine the main issues involved, and work out a solution. The answer should be a formula or algorithm for computing the dose and time interval.

It is up to you to make the problem formulation precise, to determine what additional input data you might need, to make necessary reasonable assumptions and simplifications, and to decide how to organize your work in suitable subtasks. Hint: Do not mix all difficulties at the same time. First think just on a suitable model rather than trying to solve the problem. Only consider the most basic aspects first, and make appropriate simplifications. The model can be extended later if this is needed.

Suppose that a given population can be divided into two parts: those who have a given disease and can infect others, and those who do not have it but are susceptible. Let x be the proportion of susceptible individuals and y the proportion of infectious individuals; then x + y = 1. Assume that the disease spreads by contact between sick and well members of the population and that the rate of spread dy∕dt is proportional to the number of such contacts. Further, assume that members of both groups move about freely among each other, so the number of contacts is proportional to the product of x and y. Since x=1−y, we obtain the initial value problem

dy∕dt = ?y(1 − y), y(0) = y0, (i) where ? is a positive proportionality factor, and y0 is the initial proportion of infectious individuals.

(a) Find the equilibrium points for the differential equation (i) and determine whether each is asymptotically stable, semistable, or unstable.

(b) Suppose that the equation was instead y′ = y(α − y2). Repeat your analysis from part (a). Note that your answer will depend on whether α<0, α=0, or α>0.

1. Answer the following:
   1. 145 mod 20
   2. 7¹² mod 17
2. Determine whether the following pair of numbers are relatively prime. If the answer is not, please give a reason.
   1. 34, 11
   2. 81, 15
3. Determine the following
   1. GCD(35, 10)
   2. GCD(100, 70)