Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 5.4, Problem 3E
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To justify whether it is important that the birthday be mutually independent or pairwise independent sufficient.
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Construct a truth table for the proposition and determine whether it is a contingency, a tautology, or a contradiction: (q ∨ p) ∧ ~q.
For each of the attached probabilistic expressions, please answer "yes" or "no" to indicate if it is equal to P(A,B,C), given Boolean random variables A, B and C , and no independence or conditional independence are assumed between any of them.
Construct the truth tables for the following and determine whether the compound proposition is a tautology, contradiction, or contingency.
p → (q ↔ r)
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- Draw the truth table of (A→ B) ˄ (B → A) and (A Ú B) ˄ (¬B Ú A). Determine from the truth table whether (A→ B) ˄ (B → A) is logically equivalent to ((¬A Ú B) ˄ (¬B Ú A)arrow_forwardTo do this, either show that both sides are true, or that both sides are false, for exactly the same combinations of truth values of the propositional variables in these expressions (whichever is easier). Show that (p→r)∨(q→r) and (p∧q)→r are logically equivalentarrow_forwardConstruct the truth tables for the following and determine whether the compound proposition is a tautology, contradiction, or contingency. [p ∧ (p → q) ] →qarrow_forward
- Discrete Math Answer the following:1. Show that p ∨ (q ∧ r) and (p ∨ q) ∧ (p ∨ r) are logically equivalent. This is the distributive law ofdisjunction over conjunction.2. Show that p ∨ (p ∧ q) are logically equivalent to p.3. Show that p ∧ (p ∨ q) are logically equivalent to p.4. Show that p ∨ p are logically equivalent to p.5. Show that p ∧ q are logically equivalent to p.6. Show that ¬(p → q) and p ∧ ¬q are logically equivalent.arrow_forwardShow that the following two formulas are NOT logically equivalent by giving a model in which one is true and the other is false: ∃x ( R(x) → S(x) ) and ¬ ∀x ( R(x) ∧ S(x) )arrow_forwardComplete the logical proof for the following argument. P→ (q^r) q קר . Step 1 2 3 4 5 Proposition ¬qV¬r קר Justification Hypothesis De Morgan's law, 2 Hypothesis Note that the justification for each step is either hypothesis or it would include both the name of the law or rule and the step(s) to which it is applied to.arrow_forward
- Suppose you are presented with a large integer N and are asked to find its complete factorisation. You are not told anything at all in advance about how many factors it will have, but you are instructed to use the Pollard Rho method as a probabilistic algorithm of choice. Using code snippets as may be necessary, discuss the overall struc ture of the code you would write. Show clearly where the code calls the Pollard Rho subroutine and any other sub-algorithms you may use.arrow_forward(a) Assume a probabilistic model is represented by the Bayesian network (I) in Figure 3. Is it then always possible to represent the model instead with the Bayesian network (I) of Figure 32 Given an argument for your answer. (b) Assume a probabilistic model is represented by the Markov network (III) in Figure 3. Is it then always possible to represent the model instead with the Bayesian network (1) of Figure 3? Given an argument for your answer. (1I) (II) (1)arrow_forwardDetermine whether or not the following statement is a tautology or not and give reasoning. If you need to, you can build a truth table to answer this question. (q→p)∨(∼q→∼p) A. This is a tautology because it is always true for all truth values of p and q. B. This is not a tautology because it is always false for all truth values of p and q. C. This is a tautology because it is not always false for all values of p and q. D. This is not a tautology becasue it is not always true for all truth values of p and q.arrow_forward
- Please answer the following question in depth with full detail. Consider the 8-puzzle that we discussed in class. Suppose we define a new heuristic function h3 which is the average of h1 and h2, and another heuristic function h4 which is the sum of h1 and h2. That is, for every state s ∈ S: h3(s) =h1(s) + h2(s) 2 h4(s) =h1(s) + h2(s) where h1 and h2 are defined as “the number of misplaced tiles”, and “the sum of the distances of the tiles from their goal positions”, respectively. Are h3 and h4 admissible? If admissible, compare their dominance with respect to h1 and h2, if not, provide a counterexample, i.e. a puzzle configuration where dominance does not hold.arrow_forward(a) Assume a probabilistic model is represented by the Bayesian network (I) in Figure 3. Is it then always possible to represent the model instead with the Bayesian network (I) of Figure 3? Given an argument for your answer. (b) Assume a probabilistic model is represented by the Markov network (III) in Figure 3. Is it then always possible to represent the model instead with the Bayesian network (1) of Figure 3? Given an argument for your answer. (1I) (III) B (1) Darrow_forwardIn the theoretical landscape of probability theory, a multifaceted inquiry unfolds: Can we dissect the complexities of stochastic processes and their role in modeling dynamic systems over time? How do concepts like Markov chains and Brownian motion contribute to understanding the evolution of random phenomena, transcending mere chance into a nuanced exploration of theoretical probabilities with real-world implications? Furthermore, how does the convergence of these theoretical frameworks amplify our ability to analyze and predict the uncertain trajectories of diverse phenomena in fields ranging from finance to physics?arrow_forward
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