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Discrete Math

Trees and Graphs : Vertices and Edges

1. True or False. It is possible to obtain a graph in which the number of vertices is 9, each with degree 5. 2. How many edges are there in a tree with 14 vertices? Choose one answer. (a) 10 (b) 13 (c) 14 (d) 15 (e) none of the above 3. If there 5 sections of Discrete Math with a total enrollment of 31 students, what is t

Game theory/ determining value of a game

Two DC senate candidates must decide what city to visit the day before the November election. The same four cities, Indianapolis, Evansville, Fort Wayne, and South Bend are available for both candidates. These cities are listed as strategies 1 to 4 for each candidate. Travel plans must be made in advance, so the candidates mu

two-person, zero-sum game:Value of the game

6. Consider the following two-person, zero-sum game. Identify the pure strategy. What is the value of the game? Player B ________b1_____b2_____b3 player A A1________8______5______7 A2________2______4____ _10

Proof of Splitting Field

Let K1 and K2 be finite extensions of F contained in the field K, and assume both are splitting fields over F 1. Prove that their composite K1K2 is splitting field over F. 2. Prove that K1^K1 is a splitting field over F.

Relations: reflexive, antisymmetric, transitive

For the set A = {a, b, c}, let R be the relation on A which is defined by the following 3 by 3 matrix M_R: ---------------------------------------- Row 1: 1 0 1 Row 2: 1 1 0 Row 3: 0 1 1 ----------------------------------------- Which of the properties (reflexive, antisymmetric, transitive) are satisfied by R?

Finite Math

Two cards are drawn at random from an ordinary deck of playing cards. The first is not replaced before the second is drawn. What is the probability that: a. Both cards are aces? b. At least one card is black? Using the following sample: 28, 30, 24, 30, 32, 40, 22, 25, 26, 34 a. Find the mean. b. Find the median.

Finite Math

1. Find the present value of an ordinary annuity with annual payments of $1,000, for 6 years, at 10% interest compounded annually. 2. In a marketing survey, consumers are asked to give their first three choices, of 9 different drinks. In how many different ways can they indicate their choices? 3. A class consists of 15

Compounding Interest and Value of Annuity

1. What is the ending balance from an initial deposit of $4,250 at 12% compounded quarterly for 6 years? 2. Find the present value of $5,000 in 5 years at 10% compounded annually. 3. Find the value of an annuity in which $1,100 is deposited at the end of each year for 5 years, at an interest rate of 11.5% compounded ann

Point Set Theory and Sequences

1.) Let 'S" be the set of points on the curve y = Sin (1/x) (X #0) in the XY -Plane. (a) Find a sequence of point Pn on the X - axis and in "S" such that Pn converges to (0, 0). (b) Find a sequence of pints Pn on the line Y = 1 and in "S" such that Pn coverges to (0, 1). (c) What points must be adjoined to "S" in order to ge

In a street there are 5 houses, painted 5 different colors.

1. In a street there are 5 houses, painted 5 different colors. 2. In each house lives a person of different nationality. 3. These five homeowners each drink a different kind of beverage, smoke different brand of cigar and keep a different pet. THE QUESTION: WHO OWNS THE FISH? HINTS: 1. The Brit lives in a red house

Set Partitions : Coarseness and Fineness

The meet of the partitions f1,...,fI is the finest partition that is coarser then each fi. The join of the partitions f1,...,fI is the coarsest partition that is finer than each fi. The meet of partitions is denoted and the join of partitions is denoted I

Integer Proofs by Contradiction

Assume that n is a positive integer. Use the proof by contradiction method to prove: (a) If 7n + 4 is an even integer, then n is an even integer. (b) Prove: n is an even integer iff 7n + 4 is an even integer. (Note this is an if and only if (iff) statement.

Venn Diagrams, Probability and Combinations

1. In an experiment, a pair of dice is rolled and the total number of points observed. (a) List the elements of the sample space (b) If A = { 2, 3, 4, 7, 8, 9, 10} and B = {4, 5, 6, 7, 8} list the outcomes which comprise each of the following events and also express the events in words: A, A  B, and A &#616

Can you please help me? Logic, truth tables, etc.

Please see attached file. 1. (4pts) Let p, q, and r be the following statements: p: Roses are red q: The sky is blue r: The grass is green (a) p ^ q (b) p ^ (q  r) (c) q --> (p ^ r) (d) (~ r ^ ~ q) --> ~ p 2. (4pts) Write in symbolic form using p, q, r, , , , &#

Logic Problems

1. Determine whether ~ [~ (p V ~q) <=> p V ~q. Explain the method(s) you used to determine your answer. 2. Translate the following argument into symbolic form. Determine whether the argument is valid or invalid. You may compare the form of the argument to one of the standard forms or use a truth table. If Spielberg is t

Logic

Logic 1. Let p, q, and r be the following statements: p: Roses are red q: The sky is blue r: The grass is green Translate the following statements into English (a) p &#61657; q (b) p &#61657; (q &#61658; r) (c) q &#61614; (p &#61657; r) (d) ( &#61566; r &#61657; &#61566; q) &#61614; &#61566

Venn Diagrams and Set Operations

1. In a survey of 75 consumers, 12 indicated that they were going to buy a new car, 18 said they were going to buy a new refrigerator, and 24 said they were going to buy a new washer. Of these, 6 were going to buy both a car and a refrigerator, 4 were going to buy a car and a washer, and 10 were going to buy a washer and a refri

Coding Theory : Sphere Packing and Coset Leaders

Explain what is meant by the sphere St () with centre i and radius / t in the vector space F. Show that... Let C be a linear [ri, k, dj-code over Fq and set t [i]. Show that... for all distinct elements 7 and of C. Hence show that... Give the definition of a perfect code. Give the definition of a coset leader/'Let C and t be

Coding Theory : Linear Codes

Please see the attached file for the fully formatted problems. (a) Explain what is meant by (i) a linear code over Fq (ii) the weight w(x) of a vector x (iii) the weight w(C) of a code. Prove that,... (b) Prove that w(C) = d(C) if C is a linear code. (c) Define F-linear equivalence of codes. State the three row and two

Binary Integer Programming

Consider the following activity-on-arc project network, where the 12 arcs (arrows) represent the 12 activities (tasks) that must be performed to complete the project and the network displays the order in which the activities need to be performed. The number next to each arc (arrow) is the time required for the corresponding acti

Find the bad coin out of given 8 coins using only a pan balance.

Eight coins are identical in appearance, but one coin is either heavier or lighter than the others, which all weigh the same. Describe an algorithm that identifies the bad coin in at most three weighings and also determines whether it is heavier or lighter than the others, using only a pan balance.

Continuity Proof

Let f: R-> R be a function that satisfies f(x+y) = f(x) + f(y) for all x,y in R. Suppose that f is continuous at some point c. Prove that f is continuous on R. How would you go about starting this proof?? I do not understand the f(x+y) = f(x)+f(y) thing. Does some point c make f continuous on R??

Game Theory: Matrix, Population Formulation (Lions and Lambs)

In a population, there are two kinds of individuals, LIONS and LAMBS. Whenever two individuals meet, 40 yen is at stake. When two LIONS meet, they fight each other until one of them is seriously injured. While the winner gets all the money, the loser has to pay 120 yen to get well again. If a LION meets a LAMB then the LION take

Normal subgroup proof

Let G be a finite group, let N be a normal subgroup of G, and let x be an element of G. Show that if the order of x in G is relatively prime to |G|/|N|, then x is an element of N. We know that xNx^(-1) is identical to N when N is normal, for any x. Also we know that |G|/|N| is a factor of (or divides) |G|. How to show x i

Analytic Function Proofs on Bounded Regions

(a) Let f be analytic in a bounded region D and its boundary C, such that |f(z)| = 1 on C. Show that f has at least one zero inside D, unless f is a constant. (b) Let f(z) be an analytic function in a region D except for one simple pole and assume |f(z)| = 1 on the boundary of D. Prove that every value a with |a| > 1 is take

Using MATLAB to Generate Random Numbers

This question has 3 parts: a) Write a computer program using MATLAB to generate random numbers. Use your program to generate, say, 100,000 random numbers. How long did the computer take to generate the random numbers? Roughly how long does it take for the computer to generate a single random number? b) Using a sample of th