Please written by computer source Notation:f: S -> T means f is a function with domain S and codomain TThe cardinality of the set N (the set of positive integers) is xo. (x is the firstletter in the Hebrew alphabet, pronounced "aleph".)If S and T are finite with |S| = m and |T| = n, then the number of functions f: S -> T isnm (consult a discrete structures book). This formula applies to infinite sets as well.Hence the number of functions f: N -> N is *o*o.Fact: A set with cardinality xoo is an uncountable set. [For you mathematicians, *0*0is equal to the cardinality of the power set of N, 2%o, which is also the cardinality ofthe set of real numbers.]Definition: a function f is called effectively computable if there is an algorithm tocompute the function, that is, an algorithm that, when given the value of n, allowsone to compute the value of f(n).Problem: Prove that there exist functions f: N -> N that are not effectivelycomputable. (Note - don't ask to see such a function because how could wedescribe it without trying to resort to an algorithm for the mapping?)

Cardinality: In mathematics, cardinality refers to the size or number of elements in a set. The…

Problem: Prove that there exist functions f: N -> N that are not effectively computable. (Note - don't ask to see such a function because how could we describe it without trying to resort to an algorithm for the mapping?)

Problem: Prove that there exist functions f: N -> N that are not effectively computable. (Note - don't ask to see such a function because how could we describe it without trying to resort to an algorithm for the mapping?)

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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Question

Please written by computer source

Notation:
f: S -> T means f is a function with domain S and codomain T
The cardinality of the set N (the set of positive integers) is xo. (x is the first
letter in the Hebrew alphabet, pronounced "aleph".)
If S and T are finite with |S| = m and |T| = n, then the number of functions f: S -> T is
nm (consult a discrete structures book). This formula applies to infinite sets as well.
Hence the number of functions f: N -> N is *o*o.
Fact: A set with cardinality xoo is an uncountable set. [For you mathematicians, *0*0
is equal to the cardinality of the power set of N, 2%o, which is also the cardinality of
the set of real numbers.]
Definition: a function f is called effectively computable if there is an algorithm to
compute the function, that is, an algorithm that, when given the value of n, allows
one to compute the value of f(n).
Problem: Prove that there exist functions f: N -> N that are not effectively
computable. (Note - don't ask to see such a function because how could we
describe it without trying to resort to an algorithm for the mapping?)

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