Let S be the subset of the set of ordered pairs of integers defined recursively by **Basis step**: \( (0, 0) \in S \) **Recursive step**: If \( (a, b) \in S \), then \( (a + 2, b + 3) \in S \) and \( (a + 3, b + 2) \in S \).(a) List the elements of \( S \) produced by the first five applications of the recursive definitions;(b) Use structural induction to show that \( 5|(a + b) \) when \( (a, b) \in S \).

Solution: Given: 0, 0∈S a+2, b+3 whenever a, b∈Sa+3, b+2 whenever a,…

Basis step: (0,0) = S Recursive step: If (a, b) € S, then (a +2, b+3) € S and (a +3, b+ 2) € S. (a) List the elements of S produced by the first five applications of the recursive definitions; (b) Use structural induction to show that 5|(a + b) when (a, b) € S.

Basis step: (0,0) = S Recursive step: If (a, b) € S, then (a +2, b+3) € S and (a +3, b+ 2) € S. (a) List the elements of S produced by the first five applications of the recursive definitions; (b) Use structural induction to show that 5|(a + b) when (a, b) € S.

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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Computational Systems

The computation systems can be defined as the systems that are capable of solving a problem that includes calculations either mathematical or logical, and are able to produce the result as an output. For example, a simple calculator that is given a set of numbers and operators as input and produces the result as an output can be defined as a computational system. The above example is a simple illustration only, the computational systems range from simple devices to complex devices such as the systems that control the SpaceX companies, Starlink satellites which are being quite recently seen in the skies at many places globally.

Question

Let S be the subset of the set of ordered pairs of integers defined recursively by

**Basis step**: \( (0, 0) \in S \)
**Recursive step**: If \( (a, b) \in S \), then \( (a + 2, b + 3) \in S \) and \( (a + 3, b + 2) \in S \).

(a) List the elements of \( S \) produced by the first five applications of the recursive definitions;

(b) Use structural induction to show that \( 5|(a + b) \) when \( (a, b) \in S \).

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