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Discrete Math

Proof/logic

To prove a statement "If P then Q", it is valid to prove which of the following statements instead? A. If not Q then not P B. If Q then P C. If not P then not Q D. Q only if P E. Both P and Q are true

The number of chain links required to pay for n days.

A traveler owing a gold chain with 7 links is accepted at an inn on condition that he pay one link of the chain for each day he stays. if the traveler is to pay daily and may be given links already used in payment as change, show that he only needs to take out one of the links of the chain in order to pay each day for 7 days. (n

Proof Set is Countable : Bolzano-Weierstrass Theorem

Given S is a subset of R Suppose S' (set of all accumulation points in S) = emptyset Prove S is countable. I think I am supposed to use the Bolzano-Weierstrass Theorem but I can't figure out how to apply it.

Venn Diagrams : Union and Intersection of Sets

Given the universal set of {x 0<x< 10}(this should read less than and equal too) and sets A,B,and C as defined below: A={factors of 6) B= {factors of 10} C= {odd numbers} a. List the elements in A U B U C. (they are suppossed to be upside down U"s ) b. State A U (B U C). (the U in between the b, c is the wrong way).

Probabilities and Set Theory

We say that an event A E A is nearly certain if A is nearly certainly equal to OMEGA. In other words, OMEGA = AUN , where N is a negligeable set.

Probabilities and Set Theory

Please see the attached file for the fully formatted problems. Let (Omega, A, P) be a probability space. We consider a series of mesurable sets (An)nEnCA . Prove that P(lim infnAn).... Prove that if the series is convergent, we have continuity, i.e. ...

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