2. Given the following recursive algorithm: procedure sum(x, y) Input: x: nonnegative integer, y: nonnegative integer Output: x+y: nonnegative integer if y--0 then return x else return sum(x, y-1) + 1 Prove that sum(x, y) returns x+y for any nonnegative integers x and y.
2. Given the following recursive algorithm: procedure sum(x, y) Input: x: nonnegative integer, y: nonnegative integer Output: x+y: nonnegative integer if y--0 then return x else return sum(x, y-1) + 1 Prove that sum(x, y) returns x+y for any nonnegative integers x and y.
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 2 only please.
Transcribed Image Text:2. Given the following recursive algorithm:
procedure sum(x, y)
Input: x: nonnegative integer, y: nonnegative integer
Output: x+y: nonnegative integer
if y==0 then return x
else return sum(x, y-1) + 1
Prove that sum(x, y) returns x+y for any nonnegative integers x and y.
Transcribed Image Text:Part I: Induction
Prove each of the following statements using induction, strong induction,
or structural induction. For each statement, answer the following questions.
а.
Complete the basis step of the proof.
b.
What is the inductive hypothesis?
C.
What do you need to show in the inductive step of the proof?
d.
Complete the inductive step of the proof.
1. Let S be the set of perfect binary trees, defined as:
• Basic step: a single vertex with no edges is a perfect binary tree T,.
• Recursive step: if T1 and T2 are perfect binary trees of the same height, then a new
perfect binary tree T' can be constructed by taking T1 and T2, adding a new vertex
v, and adding edges between v and the roots of T, and T2.
Prove that h(T) = log2(n(T) + 1) – 1 for any perfect binary tree T, where n(T) is the number
of vertices of T and h(T) is the height of T.
Hint: Remember that h(T,) = h(T2) and n(T,) = n(T,). Also remember that h(T') = h(T,) + 1 and
n(T') = n(T,) + n(T2) + 1.
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