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# Discrete Structures

### 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 : Reflexive, Symmetric or Transitive Relations

Let S be 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.

### 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.

### Discrete Structures: One-to-One, Onto and Range

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? See the attached file.

### Jointly Distributed Random Variables : Joint Density

The joint density of X and Y is given by... Are X and Y independent? What if f(x,y) were given by...? Please see attached.

### 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?

### Chessboard - how many paths?

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

### 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)

### Binomial Expansion Found

X +y to the seventh power

### Binomial Expansion Power

(2x + y) to the 5th power

### Subgraphs

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

### Big-Oh: Context Free Grammars

Following is a big-oh relationship. Give witnesses n0 and c that can be used to prove the relationship. Choose your witnesses to be minimal, in the sense that n0 - 1 and c are not witnesses, and if d < c, then n0 and d are not witnesses. n¹º is O(3ⁿ)

### Discrete Structures - Solving Systems of Equations

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

### Absolute Deviation with a Given Equation

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)

### Subsequences Characteristic Lines

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?

### Proving Discrete Structures

Show all steps: Prove (( n )) = (( k + 1)) (( k )) = (( n - 1))

### Inorder Traversal and Hamilton Path

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

### Trees and Rooted Binary Trees

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.

### Tree Traversal.

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

### Dimensions for Percent Increase

We have some skylights and they measure 1.2m by 0.8m Suppose both dimensions increase by 20%. What's the percent increase in the amount of light admitted? You have a newspaper the dimensions are 35cm by 38cm they reduce the pages by 10%. There are 48 Pages in the newspaper, daily circulation of 135,000. Comp

### Row operations for sets of equations

Using row operations, determine if the following set of equations has a solution: x1 + x2 + x3 = 3 2x1 - x2 - 2x3 = -3 3x2 - 4x3 = -3

### Recursive relationship problem

Consider the recursive relationship for combinations: C(n,r) = C(n-1,r) + C(n-1,r-1) Prove this relationship algebraically using the mathematical definition of a combination, as well as that of the factorial function. Provide a logical explanation for this relationship (Hint: consider n objects as consisting of n-1 ex

### Relations for Symmetric Transitive Closures

S = {0, 1, 2, 4, 6} Test the binary relations on S for reflexivity, symmetry ,antisymmetry, and transitivity. Also find the reflexive, symmetric and transitive closure of each relations. A) P = {(0,0), (1,1), (2,2), (4,4), (6,6), (0,1), (1,2), (2,4), (4,6) } B) P {(0,1), (1,0), (2,4), (4,2), (4,6), (6,4)} C) P ((0

### Statement Quantifiers Required

In the questions below suppose the variable x represents students and the variable y represents courses, and A(y): y is an advanced course S(x): x is a sophomore F(x): x is a freshman T(x,y): x is taking y. Write the statement using these predicates and any needed quantifiers. a. There is a course that every freshman is