   Chapter 9.4, Problem 38ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Observe that the sequence 12,15,8,13,7,18,19,11,14,10 has three increasing subsequences of length four: 12,15,18,19; 12,13,18,19; and 8,13,18,19. It also has one decreasing subsequence of length of n 2 + 1 distinct real numbers, there must be a sequence of length n+1 that is either strictly increasing or strictly or strictly decreasing.

To determine

To show that in any sequence of n2+1 distinct real numbers, there must be a sequence of length n+1 that is either strictly increasing or strictly decreasing.

Explanation

Given information:

Observe the sequence 12,15,8,13,7,18,19,11,14,10 has three increasing subsequence of length four; 12,15,18,19 ; 12,13,18,19 ; 8,13,18,19 it also has one decreasing subsequence of length four: 15,13,11,10.

Calculation:

Let S={a1,a2,a3,.......,an2+1} be any sequence of n2+1 distinct numbers. Let us suppose an ordered pair (mr,nr) for each ar,1rn2+1 where mr</

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