Order 7 of the following sentences so that they prove the following statement by contrapositive: For any integers m and n, it 3 + mn, then 3 + m. Proof by contrapositive of the statement (in order): Choose from this list of sentences Since 31m, there exists an integer k such that m = 3k Since k and n are integers and integers are closed under multiplication kn must be an integer 3|m Since 31mn, there exists an integer k such that mn = 3k It follows that, mn = 3kn = 3(kn) Let m and n be integers and 3 mn

Order 7 of the following sentences so that they prove the following statement by contrapositive: For any integers m and n, it 3 + mn, then 3 + m. Proof by contrapositive of the statement (in order): Choose from this list of sentences Since 31m, there exists an integer k such that m = 3k Since k and n are integers and integers are closed under multiplication kn must be an integer 3|m Since 31mn, there exists an integer k such that mn = 3k It follows that, mn = 3kn = 3(kn) Let m and n be integers and 3 mn

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Problem1Given a value `value`, if we want to make change for `value` cents, and we have infinitesupply of each of coins = {S1, S2, .. , Sm} valued `coins`, how many ways can we make the change?The order of `coins` doesn't matter.For example, for `value` = 4 and `coins` = [1, 2, 3], there are four solutions:[1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3].So output should be 4. For `value` = 10 and `coins` = [2, 5, 3, 6], there are five solutions: [2, 2, 2, 2, 2], [2, 2, 3, 3], [2, 2, 6], [2, 3, 5] and [5, 5].So the output should be 5. Time complexity: O(n * m) where n is the `value` and m is the number of `coins`Space complexity: O(n)""" def count(coins, value): """ Find number of combination of `coins` that adds upp to `value` Keyword arguments: coins -- int[] value -- int """ # initialize dp array and set base case as 1 dp_array = [1] + [0] * value.. (+.
This assignment uses several heuristic search techniques to find possibly optimal truth assignments for variables in the given Boolean formulas (see the bottom of the assignment). The formulas are in conjunctive normal form (ANDs of ORs). The fitness of an assignment is the number of clauses (ORs) that the assignment satisfies. If there are c clauses, then the highest fitness is bounded by c. However, if the formula is not satisfiable, then you cannot simultaneously make all c clauses true. You will use three of the following seven techniques: DPLL Resolution Genetic algorithms Local search Simulated annealing GSAT WalkSAT You must choose at least one complete algorithm to implement - DPLL or Resolution.
Consider nonempty set A such that JA|2 2. Specify which one of the following statements is true. * P( AxA ) n P(A) =Ø There is an injective function from A xA to A. There is a bijective function from A x A onto A. O P( AXA) n P(A) = {(0}} None of the other statements is true. O AXAS P(A)
Correct answer will be upvoted else downvoted. Computer science. You are given a network a comprising of positive integers. It has n lines and m segments. Build a lattice b comprising of positive integers. It ought to have a similar size as a, and the accompanying conditions ought to be met: 1≤bi,j≤106; bi,j is a numerous of ai,j; the outright worth of the contrast between numbers in any adjoining pair of cells (two cells that share a similar side) in b is equivalent to k4 for some integer k≥1 (k isn't really something similar for all sets, it is own for each pair). We can show that the appropriate response consistently exists. Input The primary line contains two integers n and m (2≤n,m≤500). Every one of the accompanying n lines contains m integers. The j-th integer in the I-th line is ai,j (1≤ai,j≤16). Output The output ought to contain n lines each containing m integers. The j-th integer in the I-th line ought to be bi,j.
Prove that the following problem, given a set S of integers and a number t, is of the NP class. Is there a subset of S whose elements add up to t?Note: An issue with data structures and algorithms
I need the algorithm, proof of correctness and runtime analysis for the problem. No code necessary ONLY algorithm. And runtime should be O(n log n) as stated in the question.
9. Is the following statement TRUE or FALSE? If it is TRUE, prove it, and if it is FALSE, provide a counter-example.If f: S→T is a function, where |S| > |T| (and both are finite sets), then there exist elements s1 and s2 in S such that fs1()=fs2()
3. Let p(n) = Σª a¡n², where a¿ > 0, be a degree-d polynomial in n, and let k be a constant. Prove the following properties. if k ≥ d, then p(n) = 0(nk); if k ≤d, then p(n) = N(nk); if k = d, then p(n) = 0(nk); if k > d, then p(n) = o(n²); if k < d, then p(n) = w(nk); a. b. C. d. e.
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Complete the following Pumping Lemma game to prove that L = (0 1 2-|a 262 0) is not regular. O Prover claims L is regular and fixes the value of the pumping length p. O Falsifier challenges Prover and chooses w = 0 12 L where c =. Here le| =PP O Prover writes w = xyz where y # e and sees that xy must be a string ofs only. O Let d = lylI because y #e. Falsifier now forms w' = xy'z for k =, and gets: w' = 0**1°2°L because (c + d) 0. %3D 0 123abcp|
Prove or disprove each of the following statements. To prove a statement, you should provide a formal proof that is based on the definitions of the order notations. To disprove a statement, you can either provide a counter example and explain it or provide a formal proof. All functions are positive functions. Ex.) f(n) (g(n)) ⇒ g(n) = O(f(n)) f(n) € (g(n)), for large values of n we have M₁g(n) ≤ f(n) ≤ M2g(n) for some M₁ and M₂. This means we have f(n) ≤ g(n) ≤ f(n), which shows g(n) = (f(n)). a) f(n) o(g(n)) and f(n) & w(g(n)) ⇒ f(n) = O(g(n)) b) f(n) = (h(n)) and h(n) = (g(n)) ⇒ f(n) ΕΘ(1) g(n) c) f(n) (g(n)) ⇒ 2f(n) = (29(n))
Appendix A 10-Fold Cross Validation for Parameter Selection Cross Validation is the standard method for evaluation in empirical machine learning. It can also be used for parameter selection if we make sure to use the training set only. To select parameter A of algorithm A(X) over an enumerated range d E [A1,..., A] using dataset D, we do the following: 1. Split the data D into 10 disjoint folds. 2. For each value of A e (A1,..., Ar]: (a) For i = 1 to 10 Train A(A) on all folds but ith fold Test on ith fold and record the error on fold i (b) Compute the average performance of A on the 10 folds. 3. Pick the value of A with the best average performance Now, in the above, D only includes the training data and the parameter A is chosen without the knowledge of the test data. We then re-train on the entire train set D using the chosen A and evaluate the result on the test set.