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
Program Plan Intro
To justify whether it is important that the birthday be mutually independent or pairwise independent sufficient.
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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)
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.
(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)
D
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Introduction to Algorithms
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- Show 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_forwardThis question is concerned with predicate logic in Lean Is the following proposition a tautology?(∀ x:A, PP x) → (∃ x : A, PP x)Provide a proof or a counterexample.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_forward
- Determine 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(b) Consider the statement (r = ~s) → (s = ~r): (i) Construct the truth table to the statement. (ii) Determine the statement is tautology or contradiction.arrow_forwardConsider a world with two species, emotional and unemotional. In this world emotional beings are deemed weaker than their unemotional counterparts. However, emotional beings are necessary for the overall population to survive. Using genetic algorithm, maintain a balance between these two species in such a way that the overall population is stronger, and its chances of survival are better collectively. Marks will be awarded based on the completeness, clarity and correctness of your solution.arrow_forward
- Decide which of the following formulas are valid. Justify your answer by giving a proof (in case of valid formulas) or by proposing a falsifying model (in case of invalid formulas) (a) Vx (P(x) → Q(x)) → (3x P(x)→ 3x Q(x)) (b) 3x (P(x) → Q(x)) → (Vx P(x)→ Vx Q(x)) (c) 3x (P(x)→ Q(x)) → (3x P(x)→ 3x Q(x)) (d) 3x (P(x) → Q(x)) → (Vx P(x)→ 3x Q(x)) Which of the sentences are satisfiable. Prove your answers.arrow_forwardConstruct a truth table for (p ∨ ¬ q) ∨ (¬ p ∧ q) Use the truth table that you constructed in part 1 to determine the truth value of (p ∨¬q) ∨ (¬ p ∧ q), given that p is true and q is false. Determine whether the given statement is a tautology, contradiction, or contingency. p V (~p V q) ~ (p ∧ q) ~p V ~qarrow_forwardUse Armstrong’s axioms to prove the soundness of the pseudotransitivity rule.arrow_forward
- Classify the following sentence as True or False: Reichenbach’s causation definition solves the problem of Alternative Explanations.arrow_forwardUse a truth table to determine if the following is a logical equivalence: ( q → ( ¬ q → ( p ∧ r ) ) ) ≡ ( ¬ p ∨ ¬ r )arrow_forwardPlease help explain this question into detail. I want to understand the procedure. Thank you. For each of the following, check if it is a tautology, contradiction, or neither.(a) ¬(p ∧ q) → (p → ¬q)(b) ¬(p ∧ q) → (¬p ∧ ¬q)(c) ¬(p ∧ q) ∧ p ∧ qarrow_forward
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