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Discrete Structures

Standard form

Perform operation and write the result in standard form (5 - 4i) / (3 - 2i)

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 ---

Common Solutions for Congruences

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).

Big O notation

I) Show that x^3 is O(x^4) but x^4 is not O(x^3) ii)Show that xlnx is O(x^2) but x^2 is not O(xlnx) iii)Show that a^x O(b^x) but b^x is not O(a^x) if 0 < a < b (0 = zero) iv)Show that 1^k + 2^k+...+n^k is O(n^(k+1)) for every positive integer k

Formula - 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 located

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

Bias for a random sample

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:

Frequency

When a pair of dice is rolled, the total will range from 2 (1,1) to 12 (6,6). It is a fact that some numbers will occur more frequently than others as the dice are rolled over and over. Why will some numbers come up more frequently than others? Each die has six sides numbered from 1 to 6. How many possible ways can a numb

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)

Handshakes

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 Table Functions Represented

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.

Solving Set Problems

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 ∩B).

experiment that consists of determining the type of job

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.

Discrete Structures : Coloring

Let G be a properly colored graph and let us suppose that one of the colours used is red. The set of all red-coloured vertices have a special property. What is it? Graph colouring can be thought of as partitioning V(G) into subsets with this special property. (See attachment for full background)

Subgraphs

Let G be a complete graph on n vertices. Please calculate how many spanning and induced subgroups G has... (see attachment)