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
Please see the attached file for full problem description. Let be a probability set. Prove that if is a family of events, then for all ,
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) .
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)
(A union B)* = (A*B*)* = (A* union B*)*
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.
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
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
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.
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.
Solve the following equation SEND + MORE = MONEY where each letter stands for a digit between 0 - 9 and each digit can only be used once. clues: M = 1 and O = zero
Prove that 3^(1/2) is not rational.
Show that there are no "prime triplets", that is numbers p, p+2, p+4, that are primes other than 3,5,7.
Use the euclidean algorithm to find the greatest common divisor of 981 and 1234.
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
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
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)
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?
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 asterisk 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)
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
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
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1 + sn2, with s0 = 1 and s1 = 1}. T F The First (Weak) and Second (Strong) Principles of M
I need to prove the n-cube Qn is not planar for n greater than or equal to 4.
I need to show that the number of n-bit binary strings that contain exactly two occurrences of 10 is C(n+1, 5).
Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and
Please see the attached file for the fully formatted problems. 1. Construct the truth table for the compound proposition: [p (q r)] (r p) p q r ------------------------------------------------------------- T T T T T F T F T T
Prove that G with at least (n-1)(n-2)/2+1 edges is connected, where n is the order of G.
Use the Konig-Egervary Theorem to prove that every bipartite graph G has a matching of size at e(G)/A(G) where A(G) is the maximum degree. Use this to conclude that every subgraph of Kn,n with more than (k-1)n edges has a matching at least k.
1.Use mathematical induction to prove that 2-2*7+2*7^2-.....+2(-7)^n=(1-(-7)^n+1)/4 whenever n is a nonnegative integer. 2.Show that 1^3+2^2+....n^3=[n(n+1)/2]^2 whenever n is a positive integer. 3.Use mathematical induction to show that 3 divides n^3+2n whenever n is a nonnegative integer.
Please see the attached file for the fully formatted problems. Let A = {a, b, c, d} and let the relation R be defined on A by the matrix MR = Note, take the nodes in A in the order given Use Warshall's Algorithm to determine the transitive closure of R. Draw the digraph of the transitive closure of R and use the dig