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# Discrete Math

### 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&#61602;, A &#61640; B, and A &#616

### Logic: Truth and Lies

Knight = always says the truth knave = always lying Assume there are only knight and knaves. Suppose A says: " B and C are of the same type". Then you ask C " Are A and B of the same type?" What does C answer?

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 &#61658; r) (c) q --> (p ^ r) (d) (~ r ^ ~ q) --> ~ p 2. (4pts) Write in symbolic form using p, q, r, &#61566;, &#61614;, &#61658;, &#

### 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: Truth Tables, Conditional Statements, DeMorgan's Laws and Symbolic Form

1. Write the negation for the statement below. No one in the family eats rhubarb 2. Let p, q, and r be the following statements: p: Mike is sailing q: Alice is on vacation r: Sam is in town Translate the following statement into English: (p &#61658; &#61566; q) &#61614; &#61566;r 3. Write the following co

### 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.

### Logical Operators & Truth Table

Construct a truth table for ~ p ^ (p ═ ═ > q), which is read; not p AND (p implies q)

### Encode and decode strings using given Huffman code tree.

Using the Huffman code given in the attached image, (a) encode the string "NEEDLE". (b) decode the bit string "01111001001110".

### Minimal Spanning Tree for a connected weighted graph

Find a minimal spanning tree for the connected weighted graph, following Prim's algorithm. Please see attached 4.doc for the details on graph and Prim's algorithm for finding a minimal spanning tree.

### Huffman code and other tree related problems

 Encode "LEADEN" using the Huffman code tree given in the attachment.  What can you say about a vertex in a rooted tree that has no descendants? Please see the attachment for more tree related problems.

### 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??

### Proofs

Proofs. See attached file for full problem description.

### 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

### Abelian Proof - Linear Operations

Need to figure out how to do this type of problem. Using A =[ Cos alpha - Sin alpha ] Sin alpha Cos alpha (1) Find A^-1 =[ ] E SO sub 2 (1R) (2) Check A inverse is in SO sub2 (R) Check A inverse * A = Identity and A *

### 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

### Definitions of Equivalence & Groups

(1) State the definition of equivalence relation.... and (2) Give one example of an abelian group and two (2) examples of nonabelian groups

### Definitions and Examples : Groups, Abelian Groups and Non-Abelian Groups

1. I need a simple definition of a (1) group (2) abelian group (3) nonabelian group 2. Give one example of an abelian group and 2 examples of nonabelian groups.

### Set Operations and Surjective but not Injective Mappings

Let A={-1,0,1,2} , B = {-2,3,4} and C= {-2,0,1,4}. Find: (1) (A U B) ^ C = I used ^ for "intersected with" symbol, U = union (2) (A - B) U C = (3) Give an example that a mapping from A to B that is surjective but not injective.

### Ordered Pairs and Operations on Sets

1. Recall that ordered pairs must have the property that (x,y) = (u,v) if and only if x = u and y = v. a) Prove that {{x}, {x,y}} = {{u}, {u,v}} if and only if x = u and y = v. Therefore, although we know that (x,y) does not equal {x,y} , we can define the ordered pair (x,y) as the set {{x}, {x,y}}. b) Show by an exa

### Hamiltonian proofs

A) Prove that if G and H are Hamiltonian graph, then G x H is Hamiltonian. b) Prove the n-cube Qn n>=2 is Hamiltonian.

### Sets and Set Operations

4. Let A = {a, {a}, {{a}}} B = {ø, {a}, {a, {a}}} C = {a} Be subsets of S = {ø, a, {a}, {{a}}, {a, {a}}}. Find a) A C b) B C' c) A B d) ø B e) (B C) A f) A' B g) {ø} B 5. Let A = {x | x is the name of a former president of the US} B =

### Hexadecimal to Octal to Binary Notation

Give a procedure for converting from the hexadecimal expansion of an integer to its octal expansion using binary notation as an intermediate step.

### Construction Proofs : Pentagon Proportionality and Hexagons

Please see the attached file for the fully formatted problems.