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Discrete Structures

Product of Disjoint Cycles

In S(5) let pi=(245)(1354)(125). Write pi as a product of disjoint cycles and then answer the following questions. (a) Determine pi^2, pi^5, pi^(-1). (b) What is the order of pi? Why?


1.Use the depth-first search numbering obtained in the indicated exercise to list the back edges in the graph. Use the file (5.3jpg) 2. Use Prim's algorithm to find a minimal spanning tree for each weighted graph. (Start at A) Give the weight of the minimal spanning tree found Use 5.2prims.jpg

Equivalence Relations

H is the relation on the set of all people given by H = {(a,b)|a and b are the same height}. Is H an equivalence relation? Explain your answer.


Show that ((p ══ > ~ q ) ^q) ══ > ~ p, This read (p implies not q AND q) implies not p

Standard Deviation

Given the following data set: 8, 7, 6, 9, 12 Find its standard deviation______________________

Discrete Math : Logic (40 MC Problems)

1. Identify the rule of inference used in the following: If it rains today, the flood gates will open. The flood gates did not open today. Therefore, it did not rain. a. modus tollens b. hypothetical syllogism c. modus ponens d. disjunctive syllogism 2. Identify the rule of inference used in the following: If I work all

Discrete Structures Questions

1. By using the Pigeonhole Principle, we can show that if you take six classes in a term and classes do not meet on the weekend, then at least three of the classes must meet on the same day. True False 2. By using the Pigeonhole Principle, it can be shown that if you are paid bi-weekly (every two weeks) duri

Recursive definitions

(See attached file for full problem description) --- Give a recursive definition of a) the sequence {an}, n=1,2,3,...if i. an = 1+(-1)n ii. an = n2 b) of the set of ordered pairs of positive integers S = {(a,b) | a є Z+, b є Z+, and 3 |(a+b)}.

RSA encryptions

(See attached file for full problem description) --- Consider the RSA encryption system given by p=43,q=59, and e=13 i) Find d such that ed ≡ 1 (mod (p-1)(q-1)) ii) Decode the message : 1552 2069 1178 1637 1975 Using the convention A = 00, B = 01, ..., Z = 25 ---


Find all common solutions to the congruences (in the following notations the = is meant to be a congruence symbol) x=2(mod 3), x=1(mod 4), x=3(mod 5), x=4(mod 7)

Several Problems

(See attached file for full problem description with proper symbols) --- 2. Let f(x) = x2 +1 and g(x) = {x+1, x> =3; x-1, x<3 so both f and g map R into Find the formula for a. (f+g)(x) b. (f .g)(x) c. (f o g)(x) d. (g o f)(x) 3. Let A = {a,b,c,d} and B = {1,2,3} and let f : A &#61664; B be a function . Let g : Z

Discrete structures

We worked on the attached problems today in class I am now trying to work through them again for understanding and I am not getting very far. My skills in discrete mathematics are not such that I can work through these on my own effectively. 3. Seven points are located in a plane. List the possible numbers of lines determi

Statistics - bias

If we observe a random sample X1...Xn from a distribution with the pdf... Obtain the MLE of mu. Is it unbiased? Please see attached.

Sign and rough magnitude

The plot below is the observed values of two random variables X and Y. What is the sign and a rough magnitude of the Corr (X,Y)? Please see attached.

Discrete Structures : Sequences, Subsequences and Remainders

1. How do I prove that in a group of n people there are two people with the same number of acquaintances within the group? 2. Prove that given a sequence of twelve integers, a1, a2, ...,a12, there is a subsequence aj, aj+1, ..., ak where 12 divides &#8721;kn= aa n. 3. A scrape of paper is found in an old desk that read:

Suppose that only 25% of all drivers...

48. Suppose that only 25% of all drivers come to a complete stop at an intersection having flashing red lights in all directions when no other cars are visible... (see attachment for complete question)


On the first day of math class, 20 people are present in the room. To become acquainted with one another, each person shakes hands just once with everyone else. How many handshakes take place?

Truth tables

Let p represent some false statement and let q represent some true statement. What is the truth value of each of the following: a. ~(~p v q) b. ~(p v q) c. (~p) v q

Multinomial experiment, Expected Frequency

1. True or False. In a multinomial experiment, all outcomes of each trial can have several categories. 2. In finding Expected Frequency you multiply what divided by what?

Discrete Structures : Congruences

Construct addition and multiplication tables for arithmetic modulo 11. For example, 7 + 8 mod 11 is 4 and 7*8 mod 11 is 1 so the entry in row 7, column 8 would be 4 for the addition table and 1 for the multiplication table. Use your tables to solve each of the following congruences: a. 3x+2&#8801;8 (mod11) b. 3x-5&#8801;2 (m

Discrete Structures : Reflexive, Symmetric or Transitive Relation

Let S ne the set of all strings of a's and b's. Let R be the relation on S defined by... x and y begin with different symbols. Determine, and prove your answers, whether or not R is reflexive, symmetric or transitive. Please see the attached file for the fully formatted problems.

Discrete Structures : Onto and One-to-one

Prove or disprove (find a counterexample) : If A C B and f : A --> B is an onto function (the range of f is all of B), then f is one-to-one and A =B. Please see the attached file for the fully formatted problem.

Set problem

If U = {a,b,c,d,e,f} A = {a,b} B = {-1, 0, 11} Find A' Find B' Find (A U B)' Find (A &#8745; B)

Consider an experiment that consists of determining the type of job?either blue-collar or white-collar?and the political affiliation?Republican, Democratic, or Independent?of the 15 members of an adul

16. Consider an experiment that consists of determining the type of job?either blue-collar or white-collar?and the political affiliation?Republican, Democratic, or Independent?of the 15 members of an adult soccer team. How many outcomes are (a) in the sample space; (b) in the event that at least one of the team members is a b

Application of Stirling's Formula

An often used application of Stirling's approximation is an asymptotic formula for the binomial coefficient. One can prove that for k = o(n exp3/4), (n "choose" k) ~ c(ne/k)^(k) for some appropriate constant c. Can you find the c? Can you say why this only works when k is much smaller than n exp3/4?

Double Eulerian Tour

Use words to describe the solution process. No programming. 4. Suppose G is a graph. We define a double Eulerian tour as a walk that crosses each edge of G twice in different directions and that starts and ends at the same vertex. Show that every connected graph has a double Eulerian tour.