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# Discrete Math

### 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 =

### Need to apply the Shere-Paking bound and Varshanov-Gilbert bound to know this code is linear.

Determine whether there can exist a linear binary code with the parameters [n,k,d]=[12,7,5] In here we need to apply the Shere-paking bound and Varshanov-gilbert bound to know this code is linear. I think when this code satisfies the shere-packing we do not it is a code when satisfies we know it is not a code and when it

### Singleton bound, sphere-packing bound and Varshamor-Gilbert

Can you explain what do Singleton bound, the sphere-packing bound and the Varshamor-Gilbert mean? Determine what the Singleton bound, the sphere-packing bound and the Varshamor-Gilbert bound say about the maximum number of information bits that codewords with ten check bits can have if the codewords are protected from 3 or f

### Solve: Sets and Venn Diagrams

Please view the attachment to see questions 1 and 2. 3. A survey of 10355 people restricted to those who were either female or Hispanic or over 16 years of age, produced the following data: Female: 6022 Hispanic female: 2136 Hispanic: 3564 Over 16 and female: 959 Over 16: 4722 Over 16 and Hispanic: 1341 His

### A question of group action

See attached file for full problem description.

### Proofs : GCDs and Primes

1. (i) Find the gcd (210, 48) using factorizations into primes (ii)Find (1234, 5678) 2. Prove that there are no integers x, y, and z such that x^2 + y^2 + z^2 = 999 keywords: greatest comon divisor

### Logical reasoning problem with multiple variables

It was unfortunate that Rose and the other four coworkers in her department live in different suburbs because otherwise they might have been able to carpool. As it stands, each of the five drives to work every day on a different route. Every day last week from Monday through Friday, one of the five arrived late to work because o

### Equivalence Relations and Classes

For m, n, in N define m~n if m^2 ? n^2 is a multiple of 3. (a.) Show that ~ is an equivalence relation on N. (b.) List four elements in the equivalence class [0]. c) List four elements in the equivalence class [1]. (d.) Are there any more equivalence classes. Explain your answer.

### Proof : For every positive integer n, prove that 1+2+...+n=n(n+1)/2.

Problem: For every positive integer n, prove that 1+2+...+n=n(n+1)/2.

### Game Theory : Two Person Zero Sum Game

Two armies are advancing on two cities. The first army has 4 regiments and the second army has 3 regiments. At each city, the army that send more regiments to the city captures both the city and the opposing army regiment. If both armies send the same number of regiments to a city, them the battle at the city is a draw. Each

### Real Anaylsis : Metrics Proof

Let d be a metric in X. Prove that p(x,y)=(d(x,y))/(1+d(x,y)) is also a metric in X.

### Continuity

Show that any function from a discrete metric space X into a metric space Y is continuous.

### Applications of Functions Word Problems

1. Solve the equation. 3(6 - 3x) = 1/27 2. Use natural logarithms to evaluate the logarithm to the nearest hundredth. log√4 ^259.5 3. Solve the problem. Sonja and Chris both accept new jobs on March 1, 2001. Sonja starts at \$45000 with a raise each March 1 of 3 % Chris starts at \$33000 with a raise on Mar

### Continuity Proof Intervals

Assume that f(x) is continuous in some open interval J that contains the point a, f'(x) exists for each x and limit of f'(x) as x&#61664;a exists. Prove that f is differentiable at a and f'(a)=limit of f'(x) as x&#61664;a keywords: differentiability

### Algorithm Implementation of Edge Triggered

(See attached file for full problem description) 1. S-R Latch Given the following NAND implementation of an S-R latch, Write its truth table. Qt St Rt Qt+1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 2. Gate S-R latch. Given the following implementation of a gated (clocked) S-R latch and its t

### Euler's theorem applied to large integers

Consider Euler's theorem: If m is a positive integer and a is an integer relatively prime to m, then a^phi(m)&#8801;1(mod m) Use this theorem to show that if a is an integer relatively prime to 32760 then a^12&#8801;1(mod 32760). Symbols better shown in file (attached).

### Operations Research

Infocomp Systems is a research and development laboratory firm that develops computer systems and software primarily for the medical industry. The laboratory has proposals from its own researchers for eight new projects. Each of the proposed research projects requires limited resources and it is not possible to undertake all

### Decision Theory - Expected Values

The concessions manager at our local college baseball game must decide whether to have the vendors sell sun visors or umbrellas. There is a 30% chance of rain, a 15% chance of overcast skies, and a 55% chance of sunshine, according to the weather forecast where the game is to be held. The manager estimates the following profit

### Closed-Loop Control System: Unit-Ramp Response

See the attached file. a) Using MATLAB, obtain the unit-ramp response of the closed-loop control system whose closed loop transfer function is (see attached file for equation). b) Obtain the response of this system when the input is given by r = e^-0.5t. c) Show the above inputs also along with cor

### Matlab : Lutx Function

I was trying to modify the matlab built-in lutx function, by using for loops, but when I tested the results with my new function it didn't give the same results. Please see the attached file for the fully formatted problems.

### After 100 sutdents had entered the school, which locker doors were open?

6. Students at an elementary school tried an experiment. When recess was over, each student walked into the school one at a time. The first student opened all the first 100 locker doors. The second students closed all the locker doors with even numbers. The third student changed all the locker doors with numbers that were mul

### Solving discrete math proofs

Please help with the following proofs. Answer true or false for each along with step by step proofs. 1) Prove that all integers a,b,p, with p>0 and q>0 that ((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q Or give a counterexample 2) prove for all integers a,b,p,q with p>0 and q>0 that ((a-b)mod p) mod q=0

### Discrete Mathematics and Its Applications

Key Characteristics: Please give an English text description - in your own words - highlighting key characteristics of the topic. 1. Alphabet (or vocabulary): 2. Language: 3. Type 0 grammar: 4. Derivation (or parse) tree: 5. Backus-Naur form: 6. Language recognized by an automaton: 7. Regular expression: 8. Regular se

### Prove: Set Theory, closed sets and compact sets

I would like to know how to construct a proof of union/and of 2 closed sets and how to prove compact sets. (See attached file for full problem description) --- a. Let E and F be closed sets in R. Prove that E R is closed. Prove the E F is closed. b. Let E and F be compact sets in R. Prove that E F is compact. Prove

### Distribution of intersection over union

Modern Algebra Set Theory (XVII) Laws of Algebra of Sets

### Modern Algebra, Set Theory (XI): Laws of Algebra of Sets: De Morgan's Laws (III):De Morgan's rule Complement of (A intersection B) = (Complement of A) union (Complement of B)

Modern Algebra Set Theory (XI) Laws of Algebra of Sets

### To prove the De Morgan's rule

Modern Algebra Set Theory (IX) Laws of Algebra of Sets De

### Equation Functions for Groups

6. Let (G, *) be a group. Show that each equation of either the form ax = b or the form xa = b has a unique solution in G. 7. Show that (R - {1}, *), where a * b = a + b + ab is a group