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# Discrete Math

### Proof regarding divisibility

Suppose A divides N and B divides N. Does this always imply: A * B divides n? Now the question is under what condition A*B will always divide N? Prove it.

### Relations on Set {0,1} : Binary, Reflexive and Symmetric

I have done several examples but these I cannot get right, I am not sure where I have made the mistake and I am confusing myself. a. List all the binary relations on the set {0,1}. b. List the reflexive relations on the set {0,1}. c. List the symmetric relations on the set {0,1}.

### Directed Graphs & Binary Relations

Find products A^2 & A^(2) A= [ 1 1 0 1 0 0 1 0 1 0 0 1 1 0 1 0 ]

### Trees and Their Representations

Write in prefix and infix notation: x 2 w + yz * -1

### Big-Oh Proving Question

A. Use the definition of big-oh to prove that 3n - 8 - 4n^3 / 2n - 1 is O(n^2). B. Use the definition of big-oh to prove that 1 . 2 + 2 . 3 + 3 . 4 + ... + (n - 1) . n is O(n3).

### GCD and Base 16 Factoring

A. Convert (11101)2 to base 16. b. Use the Euclidean algorithm to find gcd(34,21).

### Recursive definition

I need to give a recursive definition with initial condition(s). a.) The sequence {an}, n = 1,2,3,... where an = 2n. b.) The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, ....

### Relations

For each case, think of a set S and a binary relation p on S for - A. p is reflexive and symmetric but not transitive b. p is reflexive and transitive but not symmetric c. p is reflexive but neither symmetric nor transitive

### Characterize the Real numbers with the Discrete Metric

Characterize the set of all real numbers with the discrete metric as to whether it is compact, complete, or totally bounded. Use definitions only! (i.e. compact => every sequence converges, etc)

### Subgrpous

Prove: any subgroup of the order of p^(n-1) in a group of order p^n, where p is a prime, is a normal subgroup

### Discrete Math Assumptions

The following is is meant to have some assumptions made (like "n"). I have been up all night trying to figure this out. It can't be Euler because the vertices can't be >1. It might be Hamilton if I assume that E of G(V,E) is infinte..but how would I get my answer? I would just have sets (e1, e2,...) Could this be a straight

### Root finding of non-linear equation

Solve the equation tanh(1/x) / (1/x) = 0.95 Using the graphical, Bisection, Newton-Raphson, Regula Falsi or Muller methods. Solve the equation 1+1/[cos(x)*cosh(x)] - ax*[tan(x)-tanh(x)] Using Mueller's method

### Discrete Math : Set Relations

Let SIGMA = {a,b} be an alphabet. a. List between braces the elemnts of SIGMA4. the set of strings of length over SIGMA. b. Let A = SIGMA1 U SIGMA2 and B = SIGMA3 U SIGMA4. Describe A, B and AUB in plain English.

### Proof of Fixed Point Theorem using Stokes Theorem and Analysis

Prove that if D is the closed disc |x|  1 in R2, then any map f 2 C2[D ! D] has a fixed point: f(x) = x. The proof is by contradiction, and uses Stokes theorem. Follow the steps outlined below. (1) Define a new map F(x) = 1 .... Show that F has no fixed points if r is small enough. (2) Draw the ray from F(x) to x (these ar

### Binary Relations : Reflexive and Transitive, but not Antisymmetric

Give an example of or else prove that there are no relations on {a,b,c} that is reflexive and transitive, but not antisymmetric.

### A Discussion On Binary Relations : Reflexive, Symmetric, Antisymmetric, and/or Transitive

Determine whether the binary relation R on Z, where aRb means a^2 = b^2, is reflexive, symmetric, antisymmetric, and/or transitive.

### Game Theory: Two-Player Zero-Sum Game - Find Optimal Strategy

Please see the attached file for the fully formatted problems. Find each player's optimal strategy and the value of the two-person zero-sum game in Table 31. Player 2 Row Min Player 1 4 5 1 4 1 2 1 6 3 1 1 0 0 2 0 Column Max 4 5 6 4

### Game Theory: Two-Person Zero-Sum 2 x 3 Matrix (Mixed Strategies)

My problem is attached. I know how to work problems if the matrix is 2x2, but other sizes of matrices confuse me. Can you help? Find the value and the optimal strategies for the two person zero-sum game below. Player 2 Player 1 2 1 3 4 3 2

### Game Theory : Optimal Strategy for a 2-Person Zero-Sum Game using Mixed Strategies

Find the value and the optimal strategies for the two person zero-sum game below. Player 2 Player 1 1 2 3 2 0 3 I have determined the value of the game, but I don't know how to get to the optimal strategy. Please step through. My professor gave us the answer: Row Player Value = 4/3, The optimal strategy for the ro

### Venn Diagrams: Union, Intersection and Compliment

A. Write the first 6 elements of the following sets: E is the set of even numbers E={ } L is the set of numbers divisible by 11. L={ } S is the set of numbers divisible by 6. S={ } b. Draw a Venn Diagram to represent the relationship among E,L,S. c. Place the following five numbers on the

### Game Theory : Two-person, Constant-Sum Games

The attached file has a problem that I can't figure out how to set up. Can you take a look and explain how this problem should be set up? There are two people playing a two-person constant-sum game. Player 1 wants to travel from New York to Dallas using the shortest of the possible routes listed below. Player 2 has the ab

### Proof/logic

To prove a statement "If P then Q", it is valid to prove which of the following statements instead? A. If not Q then not P B. If Q then P C. If not P then not Q D. Q only if P E. Both P and Q are true

### Chain breaking puzzle.

A traveler owing a gold chain with 7 links is accepted at an inn on condition that he pay one link of the chain for each day he stays. if the traveler is to pay daily and may be given links already used in payment as change, show that he only needs to take out one of the links of the chain in order to pay each day for 7 days. (n

### Proof Set is Countable : Bolzano-Weierstrass Theorem

Given S is a subset of R Suppose S' (set of all accumulation points in S) = emptyset Prove S is countable. I think I am supposed to use the Bolzano-Weierstrass Theorem but I can't figure out how to apply it.

### Venn Diagrams : Union and Intersection of Sets

Given the universal set of {x 0<x< 10}(this should read less than and equal too) and sets A,B,and C as defined below: A={factors of 6) B= {factors of 10} C= {odd numbers} a. List the elements in A U B U C. (they are suppossed to be upside down U"s ) b. State A U (B U C). (the U in between the b, c is the wrong way).

### Drawing and in-circle and proof

Let r be the radius on the in-circle of tri ABC and a,b,c be the radii of the ex-circles opposite vertices A, B, and C, respectively. Illustrate the fact that 1/r = 1/a + 1/b + 1/c. Also, write a proof of this.

### Cartesian product proof

I need help constructing a proof for the Cartesian product of finitely many countable sets is countable.

### Probabilities and Set Theory..,

We say that an event A E A is nearly certain if A is nearly certainly equal to OMEGA. In other words, OMEGA = AUN , where N is a negligeable set.

### Proving Probabilities and Set Theory

Please see the attached file for the fully formatted problems. Let (Omega, A, P) be a probability space. We consider a series of mesurable sets (An)nEnCA . Prove that P(lim infnAn).... Prove that if the series is convergent, we have continuity, i.e. ...

### Probabilities and Set Theory

Please see the attached file for the fully formatted problems. Let (Omega, A) be a measurable space, and P:A--> [0,infinity] an application such that P(AUB) = P(A) + P(B) when A,B E A and A intersection B = ø, and P(Omega) = 1 . Prove that the following statements are equivalent: (i) P is a probability (ii) P is continuou