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Discrete Math

Probabilities and Set Theory

Please see the attached file for the fully formatted problems. Let (Omega, A) be a measurable space, and P:A--> [0,infinity] an application such that P(AUB) = P(A) + P(B) when A,B E A and A intersection B = ø, and P(Omega) = 1 . Prove that the following statements are equivalent: (i) P is a probability (ii) P is continuou

Probabilities : Set Theory

Please see the attached file for full problem description with proper symbols. --- Let A and B be two events such that P(A) = 3/4 and P(B) = 1/3. Prove that 1/12=<P(A intersection B)=<1/3 and give two examples where these limits are reached. In the same way, find an interval for P(AUB) .

Set Theory : Solve

If A={1,3,4}, B={2,4,6,8), C=(1,4,5} and the universe is the counting numbers less than, then find the following: A. AUB(B has line over it) B. AU(BnC)

Binary Operations : Monoids

Let S be a set with an associative binary operation but with no identity. Choose an element 1 not belonging to S, write M = {1} or S, and define an operation on M by using the operation of S and 1s=s=s1 for all s belonging to S. Show that M is a monoid.

Revenue Function, Profit Function and Maximum Profit

Problem: A company makes cameras. The price per camera at which x million cameras can be sold is: p(x) = 94.8 - 5x. 0 -< x -< 15 (the symbol -< is the "greater or equal to sign", I couldn't get it to work on my computer) The cost of making x million cameras is: c(x) = 156 + 19.7x (x is in millions of

Binary Operations : Equivalence Classes

Note. I don't how to make a letter with a line overtop of it so the equivalent notation here is *. ex) a* = a bar (a with a line overtop of it) Let M be a commutative monoid. Define a relation ~ on M by a ~ b if a = bu for some unit u. (a) Show that ~ is an equivalence on M and if a* deontes the equivalence class of a, let

Binary Operations : Idempotence

An element e of a monoid M is called an idempotent if e^2 = e. If M is finite, show that some positive power of every element is an idempotent.

Binary Operations : Cayley Table

Consider the Cayley table: (see file) Show that there is only one way to complete table (1) so that the resulting operation is associative, and that the result makes {a,b} into a commutative monoid.

Proof : Prime Triplets

Show that there are no "prime triplets", that is numbers p, p+2, p+4, that are primes other than 3,5,7.

Symbolic Logic Problem : Proof

Construct a formal proof which shows that the sentence below is a theorem of predicate logic. *the E's are existential quantifiers (usually designated by backwards E's). the & are "and". Do not use quantifier negation rules. [(x)(~Rx or Nx)& ~(Ex)Nx or (Ey)(z)Szy] ->(~(Ex)Rx or (z)(Ey)Szy

Symbolic Logic Problem : Sentence to Expression

Transcribe the English argument below into an appropriate logical language adequate to determine it to be valid. Also, please provide a derivation of the conclusion from the premises within the same logical system (by which you transcribed it). *this seems to be predicate logic and probably requires universal and existential q

Symbolic Logic : Predicate Logic

The sentence below is a theorem of predicate logic. Show that it is by deriving it from the null set of premises. If any "individual" in the domain has a property, then every individual has it. I need help explaining this and with the derivation. (EX)(FX --->(Y)FY)

Symbolic Logic : Predicate Logic

The asterisk implies a conditional usually indicated by an arrow. The & sign indicates "and". In Aristotelian logic (X)(FX*GX) logically implies (EX)(FX & GX). Is this true in predicate logic? If not, why not?

Symbolic Logic Problem

I need to know how to construct a formal proof which shows that the sentence below is a theorem of predicate logic. The ^ sign indicates the word "or". The asterics indicates a conditional usually indicated by an arrow. No quantifier negation rules can be used. [(X)(~RX^NX)&~(EX)NX^(EY)(Z)SZY] * (~(EX)RX ^ (Z)(EY)SZY)

Symbolic Logic Problem

I have to determine whether or not this formal argument below is valid. If it is I have to provide a derivation of the conclusion from the premises, which I don't know how to do. If it is invalid, an interpretation which shows the invalidity must be constructed. The & signs mean "and" usually signified by a dot. The asterisk

Discrete Math : Probability, Functional Relations, Partitions and Primary Keys

Please see the attached file for the fully formatted problems. Name ________________________________ SSN __________________ CMSC 203 - Homework Assignment 4 - Due December 9, 2003 1. (a) Suppose I have a cooler full of cans of Coke, Pepsi, Sprite, Mountain Dew, Dr. Pepper, and Slice sodas. How many distinct ways can I li

Discrete Math: Logic

Please see the attached file for the fully formatted problems. Discrete Math True or False questions 1. Circle T if the corresponding statement is True or F if it is False. T F The Fibonacci Sequence is {sn | sn = sn&#61485;1 + sn&#61485;2, with s0 = 1 and s1 = 1}. T F The First (Weak) and Second (Strong) Principles of M

Discrete Math: Logic Problems, Truth Table and Rules of Inference

Please see the attached file for the fully formatted problems. 1. Construct the truth table for the compound proposition: [p &#61658;&#61472;(&#61656;q &#61614;&#61472;&#61656;r)] &#61611;&#61472;(&#61656;r &#61614;&#61472;&#61656;p) p q r ------------------------------------------------------------- T T T T T F T F T T

Prove Connectedness

Prove that G with at least (n-1)(n-2)/2+1 edges is connected, where n is the order of G.

Discrete math

SECTION 10.5 16. Consider the “divides” relation on the following set A. Draw the Hasse diagram for the relation. (See Overview for drawing tips.) b. A = {2, 3, 4, 6, 8, 9, 12, 18} 23. Find all greatest, least, maximal, and minimal elements for the relation in #16b. 42. Use the algorithm given in the text to find a

Discrete Math: Binary Relations

Please see the attached file for the fully formatted problems. SECTION 10.2 For #2: A binary relation is defined on the set A = {0, 1, 2, 3}. For the relation given, a. draw the directed graph (See drawing tips in the Overview) b. determine whether the relation is reflexive c. determine whether the relation is symmetr

Discrete Math: Binary Relations

Please see the attached file for the fully formatted problems. 2. Let C = {2, 3, 4, 5} and D = {3, 4} and define a binary relation S from C to D as follows: for all (x, y) for all (x, y) &#61646; C &#61620; D, (x, y) &#61646; S &#61659; x &#61619; y (Yes/No answers sufficient; explanation optional) a. Is 2 S