COMP1805AB (Fall 2019)  "Discrete Structures I" Specification for Assignment 1 of 4

Please ensure that you include your name and student number on your submission. Your submission must be created using Microsoft Word, Google Docs, or LaTeX.

1. Translate the following English expressions into logical statements. You must explicitly state what the atomic propositions are (e.g., "Let p be proposition ...") and then show their logical relation.

   1. If it is red then it is not blue and it is not green.

   2. It is white but it is also red and green and blue.

   3. It is black if and only if it is not red, green, or blue.

2. Determine if the following expressions are tautologies, contradictions, or contingencies by using only truth tables. Show all your work and do not skip any steps (i.e., ensure you that you include a new column for every single operation and that you state at the end whether each expression is a tautology, contingency, or contradiction).

   a. ¬(?∧(¬?∨(?→?)))
   b. (? ∨(? ↔(¬?∧ (? ∨?))))

   c. (? ∧¬(? ∧(? ∨(? ↔?))))d. ¬(?∨¬(?→(?∧?)))

3. Determine if the following expressions are tautologies, contradictions, or contingencies by using only the logical equivalences. Show all your work and do not skip any steps (i.e., ensure you that you include the name of each equivalence used (excluding commutativity and associativity) and that you state at the end whether each expression is a tautology, contingency, or contradiction).

   a. (? ∧(? →¬?))b. (¬?→(?∨?))c. (? ↔(?∨ ¬?))d. (¬? ∨(?∧?))

4. Using only the  and the  operators, find a logical expression that is equivalent to (? → (? ∧ p)) → (? ∨ p). For this question, you do not need to specify "how" you found the equivalent expression because you will show both techniques in questions 5 and 6 below.

5. Prove that the expression you found for question 4 above is equivalent to the expression (? → (? ∧ p)) → (? ∨ p) by using only truth tables. Show all your work and do not skip any steps (i.e., ensure that you include a new column for every single operation).

6. Prove that the expression you found for question 4 above is equivalent to the expression (? → (? ∧ p)) → (? ∨ p) by using only the logical equivalences. Show all your work and do not skip any steps.

7. Let L(x) be the predicate "x is a lion", G(x) be the predicate "x is a giraffe", and M(x) be the predicate "x eats meat". Translate the following expressions into English. The universe of discourse is all animals.

   1. ∀? (?(?) ∧ ?(?))

   2. ∃? (¬(?(?) ∧ ?(?)) ∨ ¬?(?))

   3. ∀? ((?(?) ∨ ?(?)) → (¬?(?) → ?(?)))

(Investment): An investor has $150,000 to invest in oil stock, steel stock, and government bonds. The bonds are guaranteed to yield 5%, but the yield for each stock can vary. To protect against major losses, the investor decides that the amount invested in oil stock should not exceed $50,000. The total amount invested in stock CANNOT exceed the amount invested in bonds by more than $25,000.

a) Set up the problem (decision variables, problem constraints, non-negativity constraints).

b) Now form the objective function if oil stock yields 12% and the steel stock yields 9%. How much should be invested in each alternative in order to maximize the return (don't forget the bonds). What is the maximum return?

Determine if the vector(s), polynomial(s), matrices are linearly independent in R^3, P3(R), M 2x2 (R). Show algebraically how you found your answer.

a. < 2, 1, 5 > , < -2, 3, 1 > , < -4, 4, 1 >

b. x^3 - 3x^2 + 2x +1, -2x^3 + 9x^2 -3, x^3 + 6x

c. | 1 2 | | -3, -1 |

| -4  2 | , | 2 1|

(a) Find the limit of {(1/(n^(3/2)))-(3/n)+2} and use an epsilon, N argument to show that this is indeed the correct limit.

(b) Use an epsilon, N argument to show that {1/(n^(1/2))} converges to 0.

(c) Let k be a positive integer. Use an epsilon, N argument to show that {a/(n^(1/k))} converges to 0.

(d) Show that if {Xn} converges to x, then the sequence {Xn^3} converges to x^3. This has to be an epsilon, N argument [Hint: Use the difference of powers formula].

A solid is bounded by the sphere centered at the origin of radius 5 and the infinite cylinder along the z-axis of radius 3.

(a) Write inequalities that describe the solid in Cartesian coordinates.

(b) Write inequalities that describe the solid in cylindrical coordinates.

(c) Why is this solid difficult to describe in spherical coordinates? Which of the variables ρ, θ, φ are difficult to describe? Explain.

Big Red Bookstore wants to ship books from its warehouses in Brooklyn and Queens to its stores, one on Long Island and one in Manhattan. Its warehouse in Brooklyn has 700 books and its warehouse in Queens has 2,300. Each store orders 1,500 books. It costs $1 to ship each book from Brooklyn to Manhattan and $2 to ship each book from Queens to Manhattan. It costs $5 to ship each book from Brooklyn to Long Island and $4 to ship each book from Queens to Long Island.If Big Red has a transportation budget of $8,900 and is willing to spend all of it, how many books should Big Red ship from each warehouse to each store in order to fill all the orders?

Customers arrive at a local grocery store at an average rate of 2 per minute.

(a) What is the chance that no customer will arrive at the store during a given two minute period?

(b) Since it is a “Double Coupon” day at the store, approximately 70% of the customers coming to the store carry coupons. What is the probability that during a given two-minute period there are exactly four (4) customers with coupons and one (1) without coupons?

(c) Divide one given hour into 30 two-minute periods. Suppose that the numbers of customers arriving at the store during those periods are independent of each other. Denote by X the number of the periods during which exactly 5 customers arrive at the store and 4 of them carry coupons. What is the probability that X is at least 2?

(d) What is the probability that exact 4 customers coming to the store during a given two-minute period carry coupons?

Convert the follow system of equations to standard form and solve using Simplex method to find optimal solution

Maximize 10.75x + 5.3y

constraints

0.25x + 0.6y < = 1000

0.50x + 0.1y <= 1000

0.25x + 0.3y <= 1000

x>=250, y>=250

Ti Fan and his wife have retired. They are touring the USA in a big recreational vehicle (RV). They are towing a Prius car, which they use for running errands when the RV is parked. However, towing the Prius decreases the gas mileage of the RV by 25%. Ti Fan wonders if they would save gas if his wife drives the Prius instead of towing it. The Prius gets good mileage, 50 miles per gallon. Ti Fan has never wanted to know the mileage of the RV, since it would be discouraging. Nonetheless, he looks at the odometer when he fills the RV's gas tank, then looks again when he fills it again. Towing the Prius, he drove 255 miles and burned 37 gallons of gas. How much would they save, in gallons, if they had travelled the 255 miles with his wife driving the Prius while Ti Fan drove the RV?

A school is conducting optimization studies of the resources it has. One of the principal concerns of the Director is that of the staff. The problem he is currently facing is with the number of guards in the "Emergencies" section. To this end, he ordered a study to be carried out that yielded the following results:

Time Minimum number of guards required
O to 4 40
4 to 8 80
8 to 12 100
12 to 16 70
16 to 20 120
20 to 24 50

Each guard, according to Federal labor law, must work eight consecutive hours per day. Formulate the problem of hiring the minimum number of guards that meet the above requirements, as a Linear programing model. 

For the following LP problem, determine the optimal solution by the graphical solution method.

Min Z= 3x1+2x2

Subject to 2x1+x2 >10

                   -3x1+2x2 < 6

                     X1+x2 > 6

                     X1,x1 > 0

Graph and shade the feasible region

Find a polynomial of the form f(x) = ax3 + bx2 + cx + d such that f(0) = −3, f(1) = 2, f(3) = 5, and f(4) = 0. (A graphing calculator is recommended.)

answer in fraction form.