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# Discrete Math

### 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

### Partially-Ordered Sets ( Posets ) and Hom-sets

Show that a poset (partially-ordered set) is the same thing as a category in which all Hom-sets have at most one element.

### Sketch the root locus (or root loci) for the system...

Sketch the root locus (or root loci) for the system shown in Fig.1 (see attached file).

1- Prove that in a cyclic quadrilateral we have opposite angles supplementary without introducing the center of the circle. 2- Prove that a quadrilateral is paralleogram if it is simple and opposite angles are congurent.

### 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

### Bounded Proof of Functions

If {a_n + b_n} and {a_n} are both bounded, then {b_n} is bounded.

### 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.

### Compactness with two equivalent norms

(See attached file for full problem description and symbols) --- Assume that and are two equivalent norms on X, and that . Prove that M is compact in if and only if M is compact in .

### Ordered basis proof

(See attached file for full problem description with symbols) --- We have seen that the linear operator defined by is represented in the standard ordered basis by the matrix . This operator satisfies . Prove that if S is a linear operator on such that , then S = 0 or S = I, or these is an ordered basis for such that

### Linear operators proof

(See attached file for full problem description) --- Let V be a two-dimensional vector space over the field F, and let be an ordered basis for V. If is a linear operator and then prove that ---

### 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

### Extended Euclidian Algorithm Proofs

(See attached file for full problem description) --- Given positive integers a and b, the extended Euclidian algorithm constructs sequences qn, rn, sn and tn, which are defined recursively as follows: q0=0, q1=0, qn= q&#9492; rn-2/ rn-1 &#9496; for n>=2; r0=a, r1=b, rn= rn-2 - qnrn-1 for n>=2; s0=1, s1=0, sn= sn-

### 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 Proof of Divisibility

(See attached file for full problem description) --- Let d,m and n be positive integers with m>1 and m&#8801; 1 (mod d), let n= c0+mc1+m2c2+m3c3+...+mrcr be the base=m expansion of n, and let f = c0+c1+c2+c3+...+cr Prove that n is divisible by d if and only if f is divisible by d. ---

### LCM/ GCD proof

Prove for all positives integers x and y that Lcm(5x,7y) = 5* 7 * x*y ----------------------- gcd(x*gcd(5,y),7y)

### Cantor Ternary Set : Countable or Not?

How do I proof of a Cantor ternary set and how to identify whether its countable or not? (See attached file for full problem description with equation) --- Consider the set C all elements of R that have the form Where each &#945;i is either 0 or 2. Prove that in fact S is the Cantor ternary set. Given that C is the C

### 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

### Set Theory Proof

I would like to know how to prove sets involving intersection/union. (See attached file for full problem description)

### Quantificational formal logic

(See attached file for full problem description with proper symbols) --- Premise : (Ex) (Ey) (Axy v Bxy) &#61641; (Ez) (Cz) Premise : (x) (y) (Cx &#61641; ~Cy) Conclusion : /. : (x)(y)(~Axy) The E's should be backwards. I couldn't find a symbol for it. The /.: means concludes This symbol &#61641; means implies. T

### 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

### Symmetric Generalized Eigenproblem: Orthogonality Proof

For the s y m m e t r i c g e n e r a l i z e d e i g e n p r o b l e m A y =&#61472;&#61548; B y , l e t ' s p r o v e t h e o r t h o g o n a l i t y s t a t e m e n t s y JT B y i = 0 , J&#8800; i a n d yJT A y i = 0 , J &#8800; i f o r &#61548;J =&#61548; i . F i r s t , w r i

### Discrete Mathematics and Its Applications : Simplify

22. Simplify these expressions (+ = + inside a circle or 'direct sum') 1. x + 0 2. x + 1 3. x + x 4. x + x-bar For each topic, demonstrate a knowledge and capability by giving the following information: 1) Problem Solution: (solution for an even number problem) 2) Personal Observation: (personal comment

### Discrete mathematical functions

Find a transitive closure of the relation R on {a,b,c,d,e} given by R= {(a,b), (a,c), (a,e),(b,a), (b,c),(c,a), (c,b),(d,a,),(e,d)}

### Discrete Math : Show that the symmetric closure of the union of 2 relations is the union of their symmetric closures.

Show that the symmetric closure of the union of 2 relations is the union of their symmetric closures.

### Discrete Math : Functionally Complete

6. a) What does it mean for a set of operators to be functionally complete? b) Is the set {+, .} functionally complete? c) Are there sets of single operator that are functionally complete? Please see the attached file for the fully formatted problems.

### Fixed Point Iteration Problem Using Matlab

Given the function f(x) =1 + 0.5 sin x, show that it has a unique fixed point x* in the interval I=[0,2] and that the iterations xn+1=f(xn) converge to the fixed point for any x0 &#1028; I. Also, find the number of iterations necessary to guarantee that |xn - x* | < 10^-2 . Write a short Matlab code to find the fixed po