### 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)

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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)

(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  B be a function . Let g : Z

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

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.

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.

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 ∑kn= aa n. 3. A scrape of paper is found in an old desk that read:

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?

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

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?

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≡8 (mod11) b. 3x-5≡2 (m

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.

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.

Let f(x) = x^2 + x - 12. Consider f as a function from the reals to the reals. Is f one-to-one? Is f onto? What is the range of f?

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)

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

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?

The rows on a chessboard are numbered 1 to 8 and the columns have the letters a to h... (SEE ATTACHMENT)

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.

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)

X +y to the seventh power

(2x + y) to the 5th power

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

Solve the following systems of equations: (a) x=4 (5) and x=7 (11) (b) 3=34 (100) and x=-1 (51) *Please see attachment for proper symbols and complete instructions

Estimate the absolute deviation for the following calculation. List the result y and the absolute deviation. Round your answer so that it contains only significant digits. Y= 251(+/-1)*((860(+/-2))/(1.673(+/-0.006))= 129.025.70(+/-xxxxxx)

In a line of people you are looking for a subsequence of 4 (not necessarily consecutive = neighboring) people with increasing height. How many people should be in the line so that you can be sure to find this subsequence?

The attached sequences are "recursively defined" (for definition please see attachment). Please answer the questions regarding each sequence. Thank you.

Please see the attached file for the fully formatted problems. Also, can a tree have a Hamilton path?

A. If T is a rooted binary tree of height 5, then T has at most 25 leaves. b. If T is a tree with 50 vertices, the largest degree that any vertex can have is 49.

Find the: 1. preorder transversal 2. inorder transversal 3. postorder transversal Of the tree attached in the Word document.