Prove the Makeham’s second law
µx = A + Hx + BCx
Question 2
Prove the Double geometric law
µx = A + BCx + Mnx
Question 3
Prove the Perk’s law
µx =
A + BCx
KCCx + 1 + DCx
where A, B, C, D, H, K, M and n are constants.

Find expansion of

A) tan (x) about point x=0

B) cos (x) about point x=0

C) (1+x)^1/2 about point x=0

In the 1991 Gulf War, the Patriot missile defense system failed due to round off error. The troubles stemmed from a computer that performed the tracking calculations with an internal clock whose integer values in tenths of a second were converted to seconds by multiplying by a 24-bit binary approximation to one tenth:

0.1₁₀ ≈ 0.00011001100110011001100₂

(a) Convert the binary number to a decimal. Call it x.

(You may use Maple convert command: > x:=convert(0.00011001100110011001100,decimal,binary)

(b) What is the absolute error in this number; i.e., what is the absolute value of the difference between x and 0.1?

(c) What is the time error in seconds after 100 hours of operation (i.e., |3,600,000(0.1-x)|)?

(d) During the 1991 war, a Scud missile traveled at approximately MACH 5 (3750 miles per hour). Find the distance that a Scud missile would travel during the time error computed in (c).

Prove that if A is an enumerable set all of whose members are also enumerable sets, then UA is also enumerable.

Find a three digit integer in base five that has the order of its digits reversed when multiplied by 2.

Question 1

A biconditional statement whose main components are consistent statements is itself a:

		coherency
		contingency
		self-contradiction
		unable to determine from the information given
		tautology

3 points

Question 2

A biconditional statement whose main components are equivalent statements is itself a:

		self-contradiction
		coherency
		unable to determine from the information given
		contingency
		tautology

3 points

Question 3

Choose which symbol to use for “it is not the case that,” “it is false that,” and “n’t.”

		~
		•
		≡
		∨
		⊃

3 points

Question 4

A conditional statement where both the antecedent and consequent are equivalent statements is itself a:

		unable to determine from the information given
		tautology
		coherency
		contingency
		self-contradiction

3 points

Question 5

Identify which of the following is a correct symbolization of the following statement.
If the shoe fits, then one has to wear it.

		F • W
		F ≡ W
		F
		F ∨ W
		F ⊃ W

3 points

Question 6

Identify which of the following is a correct symbolization of the following statement.
      If you say it cannot be done, you should not interrupt the one doing it.

		~S ≡ ~I
		~S • ~I
		S ⊃ ~I
		~S ⊃ ~I
		~S ∨ ~I

3 points

Question 7

Identify the main connective in the following statement.

L ⊃ [(W ⊃ L) ∨ ~(Y ⊃ T)]

		~
		⊃
		≡
		∨
		•

3 points

Question 8

In the truth table for the statement form ~(p ⊃ p), the column of truth values underneath the main connective should be FF. Therefore, this statement form is a:

		contingency
		contradiction
		tautology
		equivalency
		self-contradiction

3 points

Question 9

In the truth table for the statement form p ⊃ q, the column of truth values underneath the main connective should be:

		TFFF
		TFFT
		TTTF
		TTFF
		TFTT

3 points

Question 10

In the truth table for the statement form p • q, the column of truth values underneath the main connective should be TFFF. Therefore, this statement form is a:

		tautology
		contingency
		contradiction
		equivalency
		self-contradiction

3 points

Question 11

Symbolize “both not p and not q.”

		~( p • q)
		~p • q
		( p ∨ q) • (~p ⊃ q)
		( p ∨ q) • ~( p • q)
		~p • ~q

3 points

Question 12

The connective used for biconditionals is:

		⊃
		∨
		~
		•
		≡

3 points

Question 13

The statement form p ⊂ q is:

		not actually a statement form
		a conjunction
		a conditional
		a disjunction
		a biconditional

3 points

Question 14

The following argument is an instance of one of the five equivalence rules DM, Contra, Imp, Bicon, Exp. Identify the rule.
      ~(R ⊃ U) ∨ ~(T ≡ O)
      ~[(R ⊃ U) • (T ≡ O)]

		Bicon
		DM
		Exp
		Contra
		Imp

3 points

Question 15

The following argument is an instance of one of the five equivalence rules DM, Contra, Imp, Bicon, Exp. Identify the rule.
      ~S ⊃ ~(~G ≡ U)
      (~G ≡ U) ⊃ S

		Bicon
		Exp
		DM
		Imp
		Contra

3 points

Question 16

The following argument is an instance of one of the five equivalence rules Taut, DN, Com, Assoc, Dist. Identify the rule.
      (G ∨ R) • (E ∨ S)
      [(G ∨ R) • E] ∨ [(G ∨ R) • S]

		Com
		Assoc
		DN
		Dist
		Taut

3 points

Question 17

The following argument is an instance of one of the five equivalence rules Taut, DN, Com, Assoc, Dist. Identify the rule.
      (~N ≡ D) ∨ (T • K)
      [(~N ≡ D) ∨ T] • [(~N ≡ D) ∨ K)

		Assoc
		Dist
		Taut
		Com
		DN

3 points

Question 18

The following argument is an instance of one of the five equivalence rules Taut, DN, Com, Assoc, Dist. Identify the rule.
      ~W • O
      ~~~W • O

		DN
		Com
		Assoc
		Taut
		Dist

3 points

Question 19

The following argument is an instance of one of the five inference forms MP, MT, HS, DS, Conj. Identify the form.
      [(G • R) ≡ (S ⊃ P)] ⊃ (N • G)
      ~(N • G)
      ~[(G • R) ≡ (S ⊃ P)]

		HS
		MT
		Conj
		DS
		MP

3 points

Question 20

The following argument is an instance of one of the five inference forms MP, MT, HS, DS, Conj. Identify the form.
      M ≡ O
      (M ≡ O) ⊃ (F • R)
      F • R

		MT
		DS
		MP
		HS
		Conj

3 points

Question 21

The following argument is an instance of one of the five inference forms MP, MT, HS, DS, Conj. Identify the form.
      [(P ≡ T) • (H • N)] ⊃ (T ⊃ ~S)
      (T ⊃ ~S) ⊃ [(H ∨ E) ∨ R]
      [(P ≡ T) • (H • N)] ⊃ [(H ∨ E) ∨ R]

		MP
		DS
		Conj
		MT
		HS

3 points

Question 22

The following argument is an instance of one of the five inference forms MP, MT, HS, DS, Conj. Identify the form.
      T ∨ H
~H
T

		MT
		DS
		HS
		Conj
		MP

3 points

Question 23

The following argument is an instance of one of the five inference forms MP, MT, HS, DS, Conj. Identify the form.
      (K ≡ N) ∨ (O • W)
      ~(O • W)
      (K ≡ N)

		HS
		Conj
		DS
		MT
		MP

3 points

Question 24

The following argument is an instance of one of the five inference forms Simp, Conj, Add, CD, DD. Identify the form.
      M • S
M
      M • (M • S)

		Add
		Simp
		Conj
		DD
		CD

3 points

Question 25

The following argument is an instance of one of the five inference forms Simp, Conj, Add, CD, DD. Identify the form.
      (X ⊃ M) • (R ⊃ A)
      X ∨ R
      M ∨ A

		DD
		Conj
		Add
		CD
		Simp

3 points

Question 26

The following argument is an instance of one of the five inference forms Simp, Conj, Add, CD, DD. Identify the form.
      (P ⊃ R) • (V ⊃ V)
      ~R ∨ ~V
      ~P ∨ ~V

		CD
		DD
		Simp
		Add
		Conj

3 points

Question 27

The following argument is an instance of one of the five inference forms Simp, Conj, Add, CD, DD. Identify the form.
      [(~S ≡ U) ⊃ (T ∨ E)] • [(D ∨ E) ⊃ ~N]
      (~S ≡ U) ∨ (D ∨ E)
      (T ∨ E) ∨ ~N

		DD
		Simp
		Add
		Conj
		CD

3 points

Question 28

The following argument is an instance of one of the five inference forms Simp, Conj, Add, CD, DD. Identify the form.
      [(S ∨ P) ⊃ (C ⊃ I)] • [(F ⊃ ~C) ⊃ M]
      (S ∨ P) ∨ (F ⊃ ~C)
      (C ⊃ I) ∨ M

		Simp
		CD
		DD
		Add
		Conj

3 points

Question 29

Use a short form truth table to answer the following question. Which, if any, set of truth values assigned to the atomic sentences shows that the following argument is invalid?
      A ⊃ (J ∨ S)
~J
S
A

		None—the argument is valid.
		A: F           J: F     S: T
		A: T           J: F     S: T
		A: T           J: T     S: F
		A: T           J: T     S: T

3 points

Question 30

Use a short form truth table to answer the following question. Which, if any, set of truth values assigned to the atomic sentences shows that the following argument is invalid?
      (E • ~H) ⊃ G
      ~(H ∨ G)
~E

		None—the argument is valid.
		E: T           H: F     G: T
		E: T           H: T     G: F
		E: F           H: F     G: F
		E: T           H: T     G: T

3 points

Question 31

Use a short form truth table to answer the following question. Which, if any, set of truth values assigned to the atomic sentences shows that the following argument is invalid?
      (Z ⊃ Y) ⊃ X
      Z ⊃ W
      ~Y ⊃ ~W
      V ∨ W

		Z: F           Y: F     X: T     W: F     V: F
		Z: F           Y: F     X: F     W: F     V: F
		Z: T     Y: T     X: T     W: T     V: T
		None—the argument is valid.
		Z: T           Y: T     X: F     W: F     V: F

3 points

Question 32

Use a short form truth table to answer the following question. Which, if any, set of truth values assigned to the atomic sentences shows that the following argument is invalid?
      S ⊃ R
~D
      S ⊃ D
~R

		S: F           R: F     D: F
		S: T           R: T     D: F
		S: F           R: T     D: F
		None—the argument is valid.
		S: T     R: T     D: T

3 points

Question 33

Use a short form truth table to answer the following question. Which, if any, set of truth values assigned to the atomic sentences shows that the following argument is invalid?
      (B • C) ⊃ F
      (F • E) ⊃ (J • P)
      (B • C) ⊃ P

		B: F           C: T     F: T     E: F     J: F     P: F
		B: T     C: T     F: T     E: F     J: T     P: F
		None—the argument is valid.
		B: F           C: F     F: F     E: F     J: F           P: F
		B: T           C: T     F: T     E: T     J: T     P: F

3 points

Question 34

Use a truth table to answer the following question. Which, if any, set of truth values assigned to the atomic sentences shows that the following argument is invalid?
      A ∨ B
A
~B

		A: T           B: T
		A: F           B: F
		A: F           B: T
		None—the argument is valid.
		A: T     B: F

3 points

Question 35

Use a truth table to answer the following question. Which, if any, set of truth values assigned to the atomic sentences shows that the following argument is invalid?
      ~(P • I)
      ~P ∨ ~I

		P: T           I: F
		P: F           I: F
		None—the argument is valid.
		P: T           I: T
		P: F           I: T

3 points

Question 36

Use a truth table to answer the following question. Which, if any, set of truth values assigned to the atomic sentences shows that the following argument is invalid?
      C • E
      E • C

		C: T           E: T
		C: F           E: F
		C: F           E: T
		None—the argument is valid.
		C: T           E: F

3 points

Question 37

Which rule is used in the following inference?
      (D ∨ ~E) ⊃ F
      F ⊃ (G • H)
      (D ∨ ~E) ⊃ (G • H)

		MT
		DD
		HS
		CD
		MP

3 points

Question 38

Which rule is used in the following inference?
      (A • B) ⊃ (C ⊃ D)
      A • B
      C ⊃ D

		HS
		DD
		CD
		MT
		MP

3 points

Question 39

Which rule is used in the following inference?
      [(A ⊃ B) ∨ (C ⊃ B)] ⊃ ~(~A • ~C)
      (A ⊃ B) ∨ (C ⊃ B)
      ~(~A • ~C)

		MP
		MT
		HS
		DD
		CD

3 points

Question 40

Which rule is used in the following inference?
      ~(F • K) ⊃ (F ⊃ L)
      ~(F ⊃ L)
      ~~(F • K)

		CD
		MP
		MT
		HS
		DD

3 points

Question 41

Which rule is used in the following inference?
      (B • C) ∨ D
~D
      B • C

		Conj
		Add
		Simp
		HS
		DS

3 points

Question 42

Which rule is used in the following inference?
      F ⊃ G
      ~A ∨ (F ⊃ G)

		Add
		Simp
		HS
		DS
		Conj

3 points

Question 43

Which rule is used in the following inference?
      L • ~F
~F

		Conj
		DS
		HS
		Add
		Simp

3 points

Question 44

Which rule is used in the following inference?
      E • (F ∨ G)
      H ∨ (F • G)
      [E • (F ∨ G)] • [H ∨ (F • G)]

		Conj
		DS
		HS
		Simp
		Add

3 points

Question 45

Which rule is used in the following inference?
      ~(R ∨ S) ⊃ [~O • (P ∨ Q )]
      ~(R ∨ S) ⊃ [~O • (~~P ∨ Q )]

		DN
		Assoc
		Com
		Dist
		Taut

3 points

Question 46

Which rule is used in the following inference?
      (M ≡ N) ∨ (~L • K)
      [(M ≡ N) ∨ ~L] • [(M ≡ N) ∨ K]

		Dist
		Assoc
		Taut
		Com
		DN

3 points

Question 47

Which rule is used in the following inference?
M
      M ∨ N

		Conj
		DS
		HS
		Simp
		Add

3 points

Question 48

Which, if any, of the following proofs are correct demonstrations of the validity of this argument?
      (P • Q ) • (R ∨ S)
Q
      Proof 1
(1) (P • Q ) • (R ∨ S)      /Q      Premise/Conclusion
(2) P • Q            1 Simp
(3) R ∨ S            1 Simp
(4) P                  2 Simp
(5) Q            2 Simp
      Proof 2
(1) (P • Q ) • (R ∨ S)      /Q      Premise/Conclusion
(2) P • Q            1 Simp
(3) Q            2 Simp

		Proof 2
		Proof 1
		Proofs 1 and 2
		Neither proof
		Not enough information is provided because proofs are incomplete.

3 points

Question 49

Which, if any, of the following proofs are correct demonstrations of the validity of this argument?
      (P ∨ R) ⊃ C
      C ∨ ~R
      Proof 1
(1) (P ∨ R) ⊃ C      /C ∨ ~R      Premise/Conclusion
(2) ~(P ∨ R) ∨ C            1 Imp
(3) (~P • ~R) ∨ C            2 DM
(4) C ∨ (~P • ~R)            3 Com
(5) (C ∨ ~P) • (C ∨ ~R)            4 Dist
(6) C ∨ ~R            5 Simp
      Proof 2
(1) (P ∨ R) ⊃ C      /C ∨ ~R      Premise/Conclusion
(2) ~(P ∨ R) ∨ C            1 Imp
(3) (~P • ~R) ∨ C            2 DM
(4) (~P ∨ C) • (~R ∨ C)            3 Dist
(5) ~R ∨ C            4 Simp
(6) C ∨ ~R            5 Com

		Proof 1
		Proofs 1 and 2
		Proof 2
		Not enough information is provided because proofs are incomplete.
		Neither proof

3 points

Question 50

Which, if any, of the following proofs are correct demonstrations of the validity of this argument?
      A ⊃ (B ⊃ C)
      B ⊃ (~C ⊃ ~A)
      Proof 1
(1) A ⊃ (B ⊃ C)      /B ⊃ (~C ⊃ ~A)      Premise/Conclusion
(2) (A • B) ⊃ C      1 Exp
(3) (B • A) ⊃ C      2 Com
(4) B ⊃ (A ⊃ C)      3 Exp
(5) B ⊃ (~C ⊃ ~A)      4 Contra
      Proof 2
(1) A ⊃ (B ⊃ C)      /B ⊃ (~C ⊃ ~A)      Premise/Conclusion
            (2) B      Assumption
            (3) A      Assumption
            (4) B ⊃ C      1, 3 MP
            (5) C      2, 4 MP
            (6) A ⊃ C      3–5 CP
(7) B ⊃ (A ⊃ C)      2–6 CP
(8) B ⊃ (~C ⊃ ~A)      7 Contra

		Proofs 1 and 2
		Proof 1
		Neither proof
		Proof 2
		Not enough information is provided because proofs are incomplete.

3 points

You have studied a number of mathematical structures. Vector space, metric space, topo- logical space, group, ring and field are some examples. Give general definitions and specific examples. Comment on some of the details of some of these structures. Explain how various kinds of functions are involved in these structures.

(V) Let A ⊆ R, B ⊆ R, A 6= ∅, B 6= ∅ be two bounded subset of R. Define a set A − B := {a − b : a ∈ A and b ∈ B}. Show that sup(A − B) = sup A − inf B and inf(A − B) = inf A − sup B

What are good textbooks to cover these topics?

Sets: sets and their elements, finite and infinite sets, operations on sets (unions, intersections and complements), relations between sets(inclusion, equivalence), non equivalent infinite sets, cardinal numbers.

Binary Operations: basic definitions, associativity commutativity, neutral elements, inverse elements, groups.

Functions: introduction, Cartesian products, functions as subsets of Cartesian products, graphs, composition of functions, injective, bijective and surjective functions, invertible functions, arithmetic operations on real functions, groups of functions.

Plane isometries: definition, reflections, translations and rotations, compositions of reflections, congruent triangles and isometry, classification of the plane isometries, the group of plane isometries, applications in Euclidean geometry.

Axiom systems: undefined terms, axioms and theorems of axiomatic mathematical theories, models, consistency, independence, completeness and categoricity of axiom systems, finite affine geometries.

Euclidean geometry: historical notes, a modern representation of Euclidean plane geometry as an axiomatic theory. The natural numbers: an introduction to peano's axioms,arithmetic operations, order relations, first steps in number theory, mathematical induction.

NAME: Charissa Howard

MATH125: Unit 2 Individual Project Answer Form
Number Sense, Estimation, and Financial Computations
ALL questions below regarding CONSUMER CREDIT and SAVING FOR RETIREMENT must be answered. Show ALL step-by-step calculations, round all of your final answers correctly, and include the units of measurement. Submit this modified Answer Form in the Unit 2 IP Submissions area.

CONSUMER CREDIT

For big purchases, many stores offer a deferred billing option (buy now, pay later) that allows shoppers to buy things now without paying the bill at checkout.


Assume you bought new appliances for your newly renovated home. Using the table range values below, choose one total value for the appliances that you have purchased based on the first letter of your last name. Denote this by P. It does not necessarily have to be a whole number.

First letter of your last name Possible range values for P
A–F $4,000–$4,999
G–L $5,000–$5,999
M–R $6,000–$6,999
S–Z $7,000–$7,999

Add your chosen value here:

Total value of the appliances, P $

The store where you bought these appliances offered you a provision that if you pay the bill within 3 years, you will not be charged any interest for your purchases. However, if you are even a day late in paying the bill, the store will charge you interest for the 3 years.

Choose an interest rate between 12% and 16%. Denote this by r, and convert your answer into decimal form.

Annual interest rate in decimal form, r

Suppose you forget about the bill and pay it 1 day late. How much interest do you pay if the store charges you simple interest? Because this is a dollar value, round your answer to the nearest cent. (Assume t = 3 years.)

Interest, I $

Show and explain your work here:






How much is your total bill—the total value of the appliances plus the interest? Round your answer to the nearest cent.

Total bill (simple interest) $

Show and explain your work here:






How much is your total bill if, instead, the store charges you interest that is compounded daily? Use 6 digits on your intermediate calculations, and round your final answer to the nearest cent. (Assume t = 3 years.)

Total bill (compound interest) $

Show and explain your work here:





How much interest do you pay if it is compounded daily? Round your answer to the nearest cent.

Interest, I $

Show and explain your work here:






Based on the result of your calculations, write a summary about the difference between simple and compound interest. Explain your answer.









Do you think a deferred billing option is helpful for shoppers? Explain your answer.






SAVING FOR RETIREMENT

Suppose your goal is to have a lump sum that you can withdraw when you retire. To accomplish this, you decided to contribute a portion of your paycheck to an annuity.

Using the AIU Library or the Internet, read about what kind of expenses you will be faced with when you retire. Write a brief summary of your research here:


















Based on your research, state the lump sum, in $U.S., that you want to have when you retire. This is the future value of your investment; denote it by F.

Future value, F $

State the time, in years, that you plan to contribute to your retirement account. Denote this by t.

Time, t

Based on the first letter of your last name, choose the annual interest rate for your retirement account from the chart below. It does not necessarily have to be a whole number. Denote this by r, and convert this to its decimal form.

First letter of your last name Possible values for r
A–F 5.00%–6.99%
G–L 7.00%–8.99%
M–R 9.00%–10.99%
S–Z 11.00%–12.99%

Add your chosen value here:

Annual interest rate in decimal form, r          

From the table below, choose how many times per year you want to contribute to your retirement fund. Denote this by n, and this will also be your compounding period.
Compounding Period n
Yearly 1
Semi-Annually 2
Quarterly 4
Monthly 12
Weekly 52

Add your chosen value here:

Compounding Period, n

Calculate the interest rate per compounding period, which you will denote by i, by dividing the annual interest rate from #4 by the compounding period from #5:
i=r/(n )
Round your answer to 6 decimal places.
Interest rate per compounding period, i

Show and explain your work here:





Your contribution per period, which you will denote by C, to this retirement account is calculated using the following formula:
C=(F*i)/(((1+i)^((n*t))-1) ).
Using the values that you have chosen for F, i, n, and t, calculate your contribution per period. Use 6 decimal places for your intermediate calculations, and round your final answer to the nearest cent.

Contribution amount, C $




Show and explain your work here:







Calculate your total contribution to this retirement account, which you will denote by TC, by using the formula TC = C x n x t.
Total contribution, TC $

Show and explain your work here:







What can you say about the difference in value between your total contribution (TC) and the lump sum (F) that you will receive? Based on what you have learned in this unit, is there a term that is used for this difference?

Show and explain your work here:











Summarize the results of your calculations, and explain why it is important to prepare for your retirement.

Show and explain your work here:

I need a user developed "eig" function (copy of already developed eig function from matlab) without using too much higher order math (matrix multiplication, cross and dot products and elementary row operations are all kosher) to do so, in matlab. without the use of a toolbox. if anyone can help me it would be appreciated.

Consider the function f(x) = x - xcosx, which has a root at x = 0. Write a program to compare the rates of convergence of the bisection method (starting with a = -1, b = 1) and Newton’s method (starting with x = 1). Which method converges faster? Why?

1. Prove that given n + 1 natural numbers, there are always two of them such that their difference is a multiple of n.

2. Prove that there is a natural number composed with the digits 0 and 5 and divisible by 2018.

both questions can be solved using pigeonhole principle.

Let f: X→Y be a map with A₁, A₂⊂X and B₁,B₂⊂Y

(A) Prove f(A₁∪A₂)=f(A₁)∪f(A₂).

(B) Prove f(A₁∩A₂)⊂f(A₁)∩f(A₂). Give an example in which equality fails.

(C) Prove f⁻¹(B₁∪B₂)=f⁻¹(B₁)∪f⁻¹(B₂), where f⁻¹(B)={x∈X: f(x)∈B}.

(D) Prove f⁻¹(B₁∩B₂)=f⁻¹(B₁)∩f⁻¹(B₂).

(E) Prove f⁻¹(Y∖B₁)=X∖f⁻¹(B₁).

(Abstract Algebra)

Let f: A→B and g:B→C be maps.

(A) If f and g are both one-to-one functions, show that g∘f is one-to-one.

(B) If g∘f is onto, show that g is onto.

(C) If g∘f is one-to-one, show that f is one-to-one.

(D) If g∘f is one-to-one and f is onto, show that g is one-to-one.

(E) If g∘f is onto and g is one-to-one, show that f is onto.

(Abstract Algebra)