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Discrete Math

Propositional logics

Problem: If Cleopatra was powerful, then she was venerated, but if she was not powerful, then she was not venerated and she was feared. If Cleopatra was either venerated or feared, then she was a queen. Cleopatra was a leader if she was a queen. Can you prove that Cleopatra was powerful? a leader? a queen? Do not use

List the first 10 terms of each of these sequences

Please see the attached file for the fully formatted problems. Practice problem 20 List the first 10 terms of each of these sequences. a) The sequence whose nth term is the larges integer k such that k! <= n; b) The sequence whose nth term is 3^n - 2^n; c) The sequence whose nth term is sqrt(n) ; d) The sequence whos

Discrete Math : 'Onto' Functions

To figure how many 'one-to-one' functions' there are from a set with 3 elements to a set with 4 elements --would be-- 4!/(4-3)! or 24. How do you figure out how many 'onto' functions there are from a set with 4 elements to a set with 3 elements? Please give solution and detailed explanation. Thank you.

Proof methods & strategy

Problem: Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?

Is the Set a Group?

Decide whether each of the given sets is a group with respect to the indicated operation. 1- For a fixed positive integer n, the set of all complex numbers x such that x^n=1(that is, the set of all nth roots of 1),with operation multiplication. 2-The set of all complex numbers x that have absolute value 1, with operation m

Proof : Differentiability

Suppose f(b) = f'(b) = 0 and a < b. Show that if f''(x) &#8805; 0 for x &#1028; [a,b], then f(x) &#8805; 0 for x &#1028; [a,b].

Unions and Intersections of Sets

Given A = {1, 2, 3}, B = {3, 4, 5, 6,}, and C = {3, 5, 7}. Evaluate each set a) A ∩ B b) A ∩ C c) A U C d) B U C e) (A U B) ∩ C f) A U (B U C) g) (A ∩ B) ∩ C h) (A ∩ B) U C Given the diagram below, find a) A U B and b) A ∩ B

Fractions in Binary

There are fractions in binary (floating points). Please convert 1.1 subscript 2 to decimal. Kindly show the steps. Thanks.

Euclidean Algorithm

Please see attached file for full problem description. 1. Use the euclidean algorithm to find gcd(729,75), then rerun the algorithm to find integers m and n such that gcd(729,75) = 729m + 75n. 2. Find the prime factorizations of (482,1687). Thus find the gcd and the lcm of the pair. Also find the gcd by Euclid's algorith

Trace proof

Suppose T in L(V). Prove that if trace(ST) = 0 for all S in L(V), then T = 0.

Statistics and Sampling

A survey of 100 students has the following results : 70 of the students stated they are pursuing at least one of the degrees: Mathematics, Computer Science, or Electrical Engineering. 40 were pursuing a Mathematics degree, 50 were pursuing a Computer Science degree, and 25 were pursuing an Electrical Engineering degree. 23 stu

Sets and Binary Relations : Equivalence Relations

2. Let A be the set { 1,2,3,4,5,6} and R be a binary relation on A defined as : {(1,1), (1,3), (1,5), (2,2), (2,6), (3,1), (3,3), (3,5), (4,4), (5,1), (5,3), (5,5), (6,2), (6,6)} (a) Show that R is reflexive. (b) Show that R is symmetric. (c)Show that R is transitive. 3. Let A be the set {1,2,3,4,5,6} and let F be t

Exact Sequences

If N and P are submodules of M that is an R-module and modules (N intersects P) and (N+P) are finitely generated then show that modules N and P are finitely generated.

Relations: reflexive, antisymmetric, transitive

For the set A = {a, b, c}, let R be the relation on A which is defined by the following 3 by 3 matrix M_R: ---------------------------------------- Row 1: 1 0 1 Row 2: 1 1 0 Row 3: 0 1 1 ----------------------------------------- Which of the properties (reflexive, antisymmetric, transitive) are satisfied by R?

In a street there are 5 houses, painted 5 different colors.

1. In a street there are 5 houses, painted 5 different colors. 2. In each house lives a person of different nationality. 3. These five homeowners each drink a different kind of beverage, smoke different brand of cigar and keep a different pet. THE QUESTION: WHO OWNS THE FISH? HINTS: 1. The Brit lives in a red house

Set Partitions : Coarseness and Fineness

The meet of the partitions f1,...,fI is the finest partition that is coarser then each fi. The join of the partitions f1,...,fI is the coarsest partition that is finer than each fi. The meet of partitions is denoted and the join of partitions is denoted I

Binary Bits

Part A: Use 2's complement to represent - 1910 (using 6 binary bits)? Part B: Use 2's complement representation to calculate 510 - 1910 (using 6 binary bits)? See attached file for full problem description.

Venn Diagrams, Probability and Combinations

1. In an experiment, a pair of dice is rolled and the total number of points observed. (a) List the elements of the sample space (b) If A = { 2, 3, 4, 7, 8, 9, 10} and B = {4, 5, 6, 7, 8} list the outcomes which comprise each of the following events and also express the events in words: A&#61602;, A &#61640; B, and A &#616

Logic : Truth and Lies

Knight = always says the truth knave = always lying Assume there are only knight and knaves. Suppose A says: " B and C are of the same type". Then you ask C " Are A and B of the same type?" What does C answer?

Logic Problems

1. Determine whether ~ [~ (p V ~q) <=> p V ~q. Explain the method(s) you used to determine your answer. 2. Translate the following argument into symbolic form. Determine whether the argument is valid or invalid. You may compare the form of the argument to one of the standard forms or use a truth table. If Spielberg is t

Find the bad coin out of given 8 coins using only a pan balance.

Eight coins are identical in appearance, but one coin is either heavier or lighter than the others, which all weigh the same. Describe an algorithm that identifies the bad coin in at most three weighings and also determines whether it is heavier or lighter than the others, using only a pan balance.

Truth Table

Construct a truth table for ~ p ^ (p ═ ═ > q), which is read; not p AND (p implies q)