- If B represented the set of Boolean values true and false, then the set of Boolean functions B x B → B is... A) finite B) uncountably infinite C) countably infinite - The set of functions from {0,1} to N is... A) finite B) uncountably infinite C) countably infinite - The set of functions from N to {0,1} is... A) finite B) uncountably infinite C) countably infinite

- If B represented the set of Boolean values true and false, then the set of Boolean functions B x B → B is... A) finite B) uncountably infinite C) countably infinite - The set of functions from {0,1} to N is... A) finite B) uncountably infinite C) countably infinite - The set of functions from N to {0,1} is... A) finite B) uncountably infinite C) countably infinite

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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Answer the following: This problem exercises the basic concepts of game playing, using tic-tac-toe (noughts and crosses) as an example. We define Xn as the number of rows, columns, or diagonals with exactly n X’s and no O’s. Similarly, On is the number of rows, columns, or diagonals with just n O’s. The utility function assigns +1 to any position with X3=1 and −1 to any position with O3=1. All other terminal positions have utility 0. For nonterminal positions, we use a linear evaluation function defined as Eval(s)=3X2(s)+X1(s)−(3O2(s)+O1(s)). a. Show the whole game tree starting from an empty board down to depth 2 (i.e., one X and one O on the board), taking symmetry into account. b. Mark on your tree the evaluations of all the positions at depth 2. c .Using the minimax algorithm, mark on your tree the backed-up values for the positions at depths 1 and 0, and use those values to choose the best starting move. Provide original solution!
State whether each of the following functions is injective, surjective, bijective, or none ofthese categories: (N is the set of natural numbers, and Σ∗is the set of all strings over Σ.)(a) f : N → N , f(n) = n div 3(b) g : N → N , g(n) = n2 + 2n + 1 (c) h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = a|w| (d) h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = wR
Please help me with these question. SHow all you work. Thank you 1. Prove that∀k ∈ N, 1k + 2k + · · · + nk ∈ Θ(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 ∈ Θ(g1) and f2 ∈ Θ(g2).Prove that (f1 + f2) ∈ Θ(max{g1, g2}).Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}
justify whether each of the following functions is injective, surjective, bijective, or none ofthese categories: (N is the set of natural numbers, and Σ∗ is the set of all strings over Σ.) (a) f : N → N , f(n) = n div 3(b) g : N → N , g(n) = n2 + 2n + 1 (c) h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = a|w| (d) h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = wR
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.
For each of the following functions, indicate the class (g(n)) the function belongs to. (Use the simplest g(n) possible in your answers.) Please show work to prove the assertions.
justify whether each of the following functions is injective, surjective, bijective, or none of these categories: (N is the set of natural numbers, and Σ∗ is the set of all strings over Σ.) (a) h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = wR
Use the program truth.m and a modified version of propos.m to prove or disprove the following. [(p ∧ q) ⊕ (r → s)] ⇔ [(p ∧ r) ∨ (q → s)] If it is not valid, give a counterexample; otherwise, explain why it is valid. propos.m function prop=propos(p) global num if p<0 num=3 else prop(1)=~p(1) | p(3); prop(2)=~(~p(1) | p(3)); prop(3)=p(2)&~(~p(1) | p(3)); end %Program truth.m% Construct the truth table for proposition(s) in propos.mclearformat compactglobal numpropos(-1);tot=2^num;separt=888;disp(' Basic variables 888 Proposition value(s)')for i=1:totq=i-1;for j=1:numr=rem(q,2);p(num+1-j)=r;q=(q-r)/2;endprop=propos(p);disp([p separt prop])end
Prove:Church’s thesis: The class of computable functions is equal to the class of intuitive computable functions.
5. Let R = {1, 3, π, 4, 1, 9, 10}, S = {{1}, 3, 9, 10}, T = {1, 3, π}, and U = {{1, 3, π}, 1}. Which of thefollowing are true? For those that are not, why not? (d) 1 ⊆ U(e) {1} ⊆ T(f) {1} ⊆ S(g) {1} ∈ S
Determine whether the proposed definition isa valid recursive definition of a function f from the setof nonnegative integers to the set of integers. If f is welldefined, find a formula for f(n) when n is a nonnegativeinteger and prove that your formula is valid. f(0) = 1, f(n) = −f(n − 1) for n ≥ 1
For each of the following statements, let the domain be all animals in the world. 1. Express each of the statements using quantifiers and propositional functions. 2. Form the negation of the statement so that no negation is to the left of the quantifier. 3. Express the negation in simple English. (Do not simply use the words “it is not the case that..."). (a) All dogs have fleas (b) There is a horse that can add. (c) Every koala can climb. (d)No monkey can speak French. (e) There exists a pig that can swim and catch fish.