Exercise 9 (H.W):a. Show that p^ q = 9^ p and P nd nA d r=p^ (qAb. Show that (p v 9) v r = p v (q v r) andc. Show that (p ^ 9 v r =(pn r) v (g i r)d.Show that(p v r) ^A (q v r) and (p v 9) ^ r =b - vd -=

We have to show that both left hand side and right hand side are equivalent. We can show that with…

Exercise 9 (H.W): a. Show that p n q = qa p and pv q = q v P b. Show that (p v 9) v r = pv (q v r) and (p^ 9^r = p^ (q c. Show that (p ^ 9 v r = (pへ)v(g >) d.Show that (pv r) n (9 v r) and (p v ^r = 9 ^ r = b - vd -=

Exercise 9 (H.W): a. Show that p n q = qa p and pv q = q v P b. Show that (p v 9) v r = pv (q v r) and (p^ 9^r = p^ (q c. Show that (p ^ 9 v r = (pへ)v(g >) d.Show that (pv r) n (9 v r) and (p v ^r = 9 ^ r = b - vd -=

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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4.Consider F = {BC -> D, B -> E, CE -> D, E -> CA, BF -> G} and R(A,B,C,D,E,F,G) a)Find all many keys for R
You are given a list of randomly arranged numbers, for example (11,7,18,5,17,13). The triple (11,7, 5) is called the "inversion triple" because (5<7<11) in terms of value, while the index of 5 in the list is greater than the index of 7, and the index of 7 is greater than 11. Therefore, we can find 2 inversions in such list as follows: (11,7,5), and (18,17,13). Your main task is to find the total number of inversions in any given list. a) Design a brute-force algorithm to return the number of possible inversions, and analyse the complexity of your solution b) Develop a python code to implement your brute-force algorithm. [The marks depend on the correctness of the code, indentation, comments, test-case] c) Design a more efficient algorithm to do the same task with less complexity, and analyse the complexity of your solution. [Important instruction to be followed: Create an arbitrary unsorted list of 8 characters and use it to provide full explanation of how your proposed…
Generate a random vector b of size n¡n2 × 1 and consider the function S(x) = ||Ax – b|; where || - ||2 is the vector l2 norm. Its gradient is given to be V/(x) = ATAX - ATb. Write a code to find the local minima of this function by using the gradient descent algorithm (by using the gradient expression given to you). The step size 7 in the iteration Xk+1 = X4 - 7VS(xk) should be chosen by the formula g gk gATAg where g = VS(xk) = ATAxµ – ATb. The algorithm should execute until ||Xk – Xk-1||2 < 10-4. Deliverable(s) : The python code that finds the minimum of the given function 2 and the expression for T. The values of x and f(x) should be stored (1) in a file.
PLEASE USE PYTHONGiven a jungle matrix NxM:jungle = [ [1, 0, 0, 0], [1, 1, 0, 1], [0, 1, 0, 0], [1, 1, 1, 1,]]Where 0 means the block is dead end and 1 means the block can be used in the path fromsource to destination.Task:Starting at position (0, 0), the goal is to reach position (N-1, M-1).Your program needs to build and output the solution matrix – a 4x4 matrix with 1’s inpositions used to get from the starting position (0,0) to the ending position (N-1,M-1)with the following constraints:You can only move one space at a timeYou can only in two directions: forward and down.You can only pass thru spaces on the jungle matrix marked ‘1’If you cannot reach the ending position – print a message that you’re trapped in thejungleAlgorithm:If destination is reachedprint the solution matrixElseMark current cell in the solution matrixMove forward horizontally and recursively check if this leads to a solution If there is no solution, move down and recursively check if this leads to a solution If…
For which of the following is the set S equal to {(a, b)|a, b nonnegative integers }? Select one: O A. None of these. O B. (0,0) S; (x, y) = S → (x + y, y) = S and (x,x + y) = S O C. (0,0) = S; (x, y) = S→ (x + 1, y) = S and (x, y + 1) = S O D. (0,0) E S; (x, y) = S → (x + 1, y + 1) = S
Subject/Course: Data Science.
EXM.1.AHL.TZ0.37 IB [Maximum mark: 8] In this part, marks will only be awarded if you show the correct application of the required algorithms, and show all your working. In an offshore drilling site for a large oil company, the distances between the planned wells are given below in metres. 1 2 3 4 5 6 7 8 9 10 2 30 3 40 60 4 90 190 130 5 80 200 10 160 6 70 40 20 40 40 130 7 60 120 50 90 30 60 8 50 140 90 70 70 140 70 40 9 40 170 140 60 50 90 50 70 10 200 200 80 150 110 90 30 190 90 100 11 150 30 200 120 190 120 60 190 150 200 + It is intended to construct a network of paths to connect the different wells in a way that minimises the sum of the distances between them. Use Prim's algorithm, starting at vertex 3, to find a network of paths of minimum total length that can span the whole site. [8]
If f = (log n)^2, g = n log n; what is the relationship between f and g? Choose all that applies. a. g= O(f) b. f = O(g) c. g= Ω(f) d. f = Ω(g)
An operation ∗ on the set of positive integers is defined by a ∗ b = (a + b) a−b . Evaluate 1024 ∗ (512 ∗ (256 ∗ (128 ∗ (64 ∗ (32 ∗ (16 ∗ (8 ∗ (4 ∗ (2 ∗ 1). a. 2028 b. 2048 c. 2047 d. 2027
Suppose you are in Canada's Thousand Islands National Park, and in one particular lake there are n small islands that park officials want to connect with floating bridges so that people can experience going between islands without a canoe. The cost of constructing a bridge is proportional to its length. Assume the distance between every pair of islands is given to you as a two dimensional matrix (an example of such a table for n = 8 islands is shown below). A B C Ꭰ E F G H A 240 210 340 280 200 345 120 B 240 265 175 215 180 185 155 C 210 265 - 260 115 350 435 195 Ꭰ 340 175 260 160 330 295 230 E 280 215 115 160 360 400 170 F 200 180 350 330 360 175 205 G 345 185 435 295 400 175 305 H 120 155 195 230 170 205 305 Design an algorithm for determining which bridges they should build to connect the islands at minimal cost. Write down the pseudocode and explain why your algorithm correctly computes the set of bridges of minimal cost. Analyze the runnning time of your algorithm.
Type in Latex **Problem**. Let $$A = \begin{bmatrix} .5 & .2 & .3 \\ .3 & .8 & .3 \\ .2 & 0 & .4 \end{bmatrix}.$$ This matrix is an example of a **stochastic matrix**: its column sums are all equal to 1. The vectors $$\mathbf{v}_1 = \begin{bmatrix} .3 \\ .6 \\ .1 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}$$ are all eigenvectors of $A$. * Compute $\left[\begin{array}{rrr} 1 & 1 & 1 \end{array}\right]\cdot\mathbf{x}_0$ and deduce that $c_1 = 1$.* Finally, let $\mathbf{x}_k = A^k \mathbf{x}_0$. Show that $\mathbf{x}_k \longrightarrow \mathbf{v}_1$ as $k$ goes to infinity. (The vector $\mathbf{v}_1$ is called a **steady-state vector** for $A.$) **Solution**. To prove that $c_1 = 1$, we first left-multiply both sides of the above equation by $[1 \, 1\, 1]$ and then simplify both sides:$$\begin{aligned}[1 \, 1\, 1]\mathbf{x}_0 &= [1 \, 1\, 1](c_1\mathbf{v}_1 +…
Consider an n by n matrix, where each of the n2 entries is a positive integer. If the entries in this matrix are unsorted, then determining whether a target number t appears in the matrix can only be done by searching through each of the n2 entries. Thus, any search algorithm has a running time of O(n²). However, suppose you know that this n by n matrix satisfies the following properties: • Integers in each row increase from left to right. • Integers in each column increase from top to bottom. An example of such a matrix is presented below, for n=5. 4 7 11 15 2 5 8 12 19 3 6 9 16 22 10 13 14 17 24 1 18 21 23 | 26 | 30 Here is a bold claim: if the n by n matrix satisfies these two properties, then there exists an O(n) algorithm to determine whether a target number t appears in this matrix. Determine whether this statement is TRUE or FALSE. If the statement is TRUE, describe your algorithm and explain why your algorithm runs in O(n) time. If the statement is FALSE, clearly explain why no…