B. Prove the following by using Chain of Equivalence1. {[(P V Q) A (P V ~Q)] V Q} + (P V Q)2. (P → Q) A [~Q ^ (R V ~Q)] → ~(Q V P)

Proved the given using chain of equivalence

Answered: B. Prove the following by using Chain…

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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Given a set of n positive integers, C = {c1,c2, ..., cn} and a positive integer K, is there a subset of C whose elements sum to K? A dynamic program for solving this problem uses a 2-dimensional Boolean table T, with n rows and k + 1 columns. T[i,j] 1≤ i ≤ n, 0 ≤ j ≤ K, is TRUE if and only if there is a subset of C = {c1,c2, ..., ci} whose elements sum to j. Which of the following is valid for 2 ≤ i ≤ n, ci ≤ j ≤ K? a) T[i, j] = ( T[i − 1, j] or T[i, j − ci]) b) T[i, j] = ( T[i − 1, j] and T[i, j − ci ]) c) T[i, j] = ( T[i − 1, j] or T[i − 1, j − ci ]) d) T[i, j] = ( T[i − 1, j] and T[i − 1, j − cj ]) In the above problem, which entry of the table T, if TRUE, implies that there is a subset whose elements sum to K? a) T[1, K + 1] b) T[n, K] c) T[n, 0] d) T[n, K + 1]
Given a set of n positive integers, C = {c1,c2, ..., cn} and a positive integer K, is there a subset of C whose elements sum to K? A dynamic program for solving this problem uses a 2-dimensional Boolean table T, with n rows and k + 1 columns. T[i,j] 1≤ i ≤ n, 0 ≤ j ≤ K, is TRUE if and only if there is a subset of C = {c1,c2, ..., ci} whose elements sum to j. Which of the following is valid for 2 ≤ i ≤ n, ci ≤ j ≤ K? a) ?[?, ?] = ( ?[? − 1, ?] ?? ?[?, ? − ?? ]) b) ?[?, ?] = ( ?[? − 1, ?] ??? ?[?, ? − ?? ]) c) ?[?, ?] = ( ?[? − 1, ?] ?? ?[? − 1, ? − ?? ]) d) ?[?, ?] = ( ?[? − 1, ?] ??? ?[? − 1, ? − ?? ]) In the above problem, which entry of the table T, if TRUE, implies that there is a subset whose elements sum to K? a) ?[1, ? + 1] b) ?[?, ?] c) ?[?, 0] d) ?[?, ? + 1]
Show that ((p→q) v (~(p ^ ~q) ^ T)) ≡ ~p v q using the logical equivalences.
Consider a Diffie-Hellman scheme with a common prime q = 17 and a primitive root α = 3. a) If user A has a private key XA=4, what is A’s public key, YA? b) A sends YA to B. If B has a private key XB=6, what is the shared secret key, K that B can calculate and share with A? c) If B computes YB and sends it to A, what is the shared secret Key, K computed by A?
prove the equivalence of the schemata:~(~(p • ~ q ) • ~ ( ~ p• q)) equal to ~(p • q) • ~(~p • ~q)
1. Here is a model M:Domain: {1, 2, 3, 4, 5}P : {1, 3, 5}, Q : {2, 4, 5}, R : ∅, S : {3, 4}a : 3, b : 4Is the proposition ∀x((P x ∧ Qx) → Sx) ∨ (Rb ↔ P a) true or false in M? Explain.
For the traversal log: {X, Y, Z, W, Y, A, B, C, D, Y, C, D, E, F, D, E, X, Y, A, B, M, N}, a. Find maximal forward references. b. Find large reference sequences if the threshold value is 0.3 (or 30%). c. Find maximal reference sequences.
6. Let M = (Q,Sigma,s, q0, F) be a dfa and define cfg g= (v,sigma, R,S) as follows: 1. V=Q; 2. For each q in Q and a in sigma, define rule q->aq' where q' = s(q,a); 3. S = q0 Prove L(M) = L(G)
4. Show that (p → q) ∧ (p → r) and p → (q ∧ r) are logically equivalent using equivalencelaw.
6. Consider a binary classification problem using 1-nearest neighbors with the Euclidean distance metric. We have N 1-dimensional training points x(1), x(2), . . . x(N ) and corresponding labelsy(1), y(2), . . . y(N ) with x(i ) ∈ R and y(i ) ∈ {0, 1}. Assume the points x(1), x(2), . . . x(N ) are in ascending order by value. If there are ties during the 1-NN algorithm, we break ties by choosing the labelcorresponding to the x(i ) with lower value.
Given set A = {a, b, c, d} show the equivalence relation, which contains eight ordered pairs, that induces this partition of A : {{a, c}, {b, d}}.
Consider the following knowledge base Prove that Q is true with: 1. P → Q 2. L ∧ M → P 3. B ∧ L → M 4. A ∧ P → L 5. A ∧ B → L 6. A 7. B Forward-Chaining Backward-Chaining Resolution Prove t → s: 1. p → q 2. [q ∧ r] → s 3. [t ∧ u] → r 4. u → w 5. t → y 6. y → u 7. r → p 8. p → m i. Express in clause form ii. Forward-Chaining iii. Backward-Chaining iv. Resolution

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B. Prove the following by using Chain of Equivalence 1. {[(P V Q) ^ (P V ~Q)] V Q} + (P V Q) 2. (P- Q) A [~Q ^ (R V ~Q)] ++ ~(Q V P)