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Discrete Math

Binary tree

A. Write 3n − (k + 5) in prefix notation: ????. b. If T is a binary tree with 100 vertices, its minimum height is ????. c. Every full binary tree with 50 leaves has ???? vertices.


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}.

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).


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, ....


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


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

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

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.

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 divisble 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 V

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


To prove that a statement of the form "If P then Q", you may assume that: A. P=Q B. P is true C. Q is true D. P is false E. nothing until it has been proved


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

The number of chain links required to pay for n days.

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).

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.

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

Probabilities : Set Theory

Please see the attached file for full problem description with proper symbols. --- Let A and B be two events such that P(A) = 3/4 and P(B) = 1/3. Prove that 1/12=<P(A intersection B)=<1/3 and give two examples where these limits are reached. In the same way, find an interval for P(AUB) .