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Discrete Math

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.

Game Theory: Matrix, Population Formulation (Lions and Lambs)

In a population, there are two kinds of individuals, LIONS and LAMBS. Whenever two individuals meet, 40 yen is at stake. When two LIONS meet, they fight each other until one of them is seriously injured. While the winner gets all the money, the loser has to pay 120 yen to get well again. If a LION meets a LAMB then the LION take

Ordered Pairs and Operations on Sets

1. Recall that ordered pairs must have the property that (x,y) = (u,v) if and only if x = u and y = v. a) Prove that {{x}, {x,y}} = {{u}, {u,v}} if and only if x = u and y = v. Therefore, although we know that (x,y) does not equal {x,y} , we can define the ordered pair (x,y) as the set {{x}, {x,y}}. b) Show by an exa

Sets and Set Operations

4. Let A = {a, {a}, {{a}}} B = {ø, {a}, {a, {a}}} C = {a} Be subsets of S = {ø, a, {a}, {{a}}, {a, {a}}}. Find a) A C b) B C' c) A B d) ø B e) (B C) A f) A' B g) {ø} B 5. Let A = {x | x is the name of a former president of the US} B =

Singleton bound, sphere-packing bound and Varshamor-Gilbert

Can you explain what do Singleton bound, the sphere-packing bound and the Varshamor-Gilbert mean? Determine what the Singleton bound, the sphere-packing bound and the Varshamor-Gilbert bound say about the maximum number of information bits that codewords with ten check bits can have if the codewords are protected from 3 or f

Truth Values : Predicates and Quantifiers

Determine the truth value of the statement $x"y(x<= y²) (and explain your answer) if the universe of discourse for the variables consists of: a) the positive real numbers b) the integers c) the nonzero real numbers

Solve: Sets and Venn Diagrams

Please view the attachment to see questions 1 and 2. 3. A survey of 10355 people restricted to those who were either female or Hispanic or over 16 years of age, produced the following data: Female: 6022 Hispanic female: 2136 Hispanic: 3564 Over 16 and female: 959 Over 16: 4722 Over 16 and Hispanic: 1341 His

Proofs : GCDs and Primes

1. (i) Find the gcd (210, 48) using factorizations into primes (ii)Find (1234, 5678) 2. Prove that there are no integers x, y, and z such that x^2 + y^2 + z^2 = 999 keywords: greatest comon divisor

Linear Combinations, Division and the Euclidean Algorithm

Assume that d=sa+tb is a linear combination of integers a and b. Find infinitely many pairs of integers ( s sub k, t sub k ) with d=s sub k a + t sub k b Hint: If 2s +3t =1, then 2 (s+3) + 3 (t-2) = 1 I would like a very detailed, as possible, exp

Equivalence Relations and Classes

For m, n, in N define m~n if m^2 ? n^2 is a multiple of 3. (a.) Show that ~ is an equivalence relation on N. (b.) List four elements in the equivalence class [0]. c) List four elements in the equivalence class [1]. (d.) Are there any more equivalence classes. Explain your answer.

Mod and Divisibility Proof

Let i, j, n be positive integers with i > j. Let f(x) in Zn[x] have non-zero constant term, and let d = o(x mod f(x)). Suppose that x^i and x^j have the same remainder on division by f(x). Prove that i-j >= d. -- Theorem 4.7.2 Suppose that F is a field of order |F| = q and that f(x) in F[x] has degree n >= 1 and has non-zer

Proof : Intergers Modulo n

Prove that if a-bar,b-bar &#1028; (Z/nZ)^x then a-bar.b-bar &#1028; (Z/nZ)^x Please see the attached file for the fully formatted problems.

Construction and proof involving line segments.

Given a line segment QR=1 1. Given segments qand r, constuct a line segment qr. 2. Given segments q,r,s construct a line segment qrs. I'm confused on how to set it up and begin. I have the one for constructing a line segment of length q/r. Thanks

Drawing Venn Diagrams

Make a venn diagram: 25 students play soccer, 4 boys play soccer and baseball, 3 girls play soccer and baseball, 10 boys play baseball, 4 girls play baseball, 3 boys plays baseball and tennis, 1 girl plays baseball and tennis, 1 boy plays all three sports, 1 girl plays all three sports, 9 students play tennis, 3 boys play soccer

Game Theory : Two Person Zero Sum Game

Two armies are advancing on two cities. The first army has 4 regiments and the second army has 3 regiments. At each city, the army that send more regiments to the city captures both the city and the opposing army regiment. If both armies send the same number of regiments to a city, them the battle at the city is a draw. Each

Two-Person Zero-Sum Game Problem

Need help setting up this 3X4 Matrix problem. I have only worked 2X2 matrix game problems per recent homework, however, I can't seem to leverage that knowledge for this problem. I'm also not sure how to deal with the extra reward variable (one point for capturing each of the other army's bn). Also need help setting up the LP for


Show that any function from a discrete metric space X into a metric space Y is continuous.

Applications of Functions Word Problems

1. Solve the equation. 3(6 - 3x) = 1/27 2. Use natural logarithms to evaluate the logarithm to the nearest hundredth. log√4 ^259.5 3. Solve the problem. Sonja and Chris both accept new jobs on March 1, 2001. Sonja starts at $45000 with a raise each March 1 of 3 % Chris starts at $33000 with a raise on Mar