2. Let S be a sequence of n different numbers. A subsequence of S is a sequence that can be obtained by deleting elements of S. For example, if S is (6, 4, 7, 9, 1, 2, 5, 3, 8), then (6, 4, 7) and (7,2,5, 3) are both sub- sequences of S. An increasing subsequence of S is a subsequence of whose successive elements get larger. For example, (1, 2, 3, 8) is an increasing subsequence of S. Decreasing subse- quences are defined similarly; (6, 4, 1) is a decreasing subsequence of S. And let A be the set of numbers in S. (So A is [1,9] for the example above.) There are two straightforward linear orders for A. The first is numerical order where A is ordered by the relation. The second is to order the elements by which comes first in S; call this order
2. Let S be a sequence of n different numbers. A subsequence of S is a sequence that can be obtained by deleting elements of S. For example, if S is (6, 4, 7, 9, 1, 2, 5, 3, 8), then (6, 4, 7) and (7,2,5, 3) are both sub- sequences of S. An increasing subsequence of S is a subsequence of whose successive elements get larger. For example, (1, 2, 3, 8) is an increasing subsequence of S. Decreasing subse- quences are defined similarly; (6, 4, 1) is a decreasing subsequence of S. And let A be the set of numbers in S. (So A is [1,9] for the example above.) There are two straightforward linear orders for A. The first is numerical order where A is ordered by the relation. The second is to order the elements by which comes first in S; call this order
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:(d) Prove that every sequence S of length n has an increasing subsequence of length
greater than n or a decreasing subsequence of length at least vn.
![2. Let S be a sequence of n different numbers. A subsequence of S is a sequence that
can be obtained by deleting elements of S.
For example, if S is (6, 4, 7,9, 1, 2, 5, 3, 8), then (6, 4, 7) and (7, 2, 5, 3) are both sub-
sequences of S.
An increasing subsequence of S is a subsequence of whose successive elements get
larger. For example, (1, 2,3, 8) is an increasing subsequence of S. Decreasing subse-
quences are defined similarly; (6, 4, 1) is a decreasing subsequence of S.
And let A be the set of numbers in S. (So A is [1,9] for the example above.) There
are two straightforward linear orders for A. The first is numerical order where A is
ordered by the < relation. The second is to order the elements by which comes first
in S; call this order <s. So for the example above, we would have
6 <s 4 <s 7 <s 9 <s 1 <s 2 <s 5 <s 3 <s 8.
Let < be the product relation of the linear orders <s and <. That is, < is defined
by the rule
a < a' := a < a' and a <s a'.
So < is a partial order on A.
(a) List all the maximum-length increasing subsequences of S, and all the maximum-
length decreasing subsequences.
(b) Draw a diagram of the partial order < on A. What are the maximal and minimal
elements?
(c) Explain the connection between increasing and decreasing subsequences of S,
and chains and anti-chains under <.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9be273bd-a490-45b4-94b1-b8a2b1c66413%2Fdaf720e8-d3c4-473c-9bde-875a03ed0ce5%2F08x0uh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let S be a sequence of n different numbers. A subsequence of S is a sequence that
can be obtained by deleting elements of S.
For example, if S is (6, 4, 7,9, 1, 2, 5, 3, 8), then (6, 4, 7) and (7, 2, 5, 3) are both sub-
sequences of S.
An increasing subsequence of S is a subsequence of whose successive elements get
larger. For example, (1, 2,3, 8) is an increasing subsequence of S. Decreasing subse-
quences are defined similarly; (6, 4, 1) is a decreasing subsequence of S.
And let A be the set of numbers in S. (So A is [1,9] for the example above.) There
are two straightforward linear orders for A. The first is numerical order where A is
ordered by the < relation. The second is to order the elements by which comes first
in S; call this order <s. So for the example above, we would have
6 <s 4 <s 7 <s 9 <s 1 <s 2 <s 5 <s 3 <s 8.
Let < be the product relation of the linear orders <s and <. That is, < is defined
by the rule
a < a' := a < a' and a <s a'.
So < is a partial order on A.
(a) List all the maximum-length increasing subsequences of S, and all the maximum-
length decreasing subsequences.
(b) Draw a diagram of the partial order < on A. What are the maximal and minimal
elements?
(c) Explain the connection between increasing and decreasing subsequences of S,
and chains and anti-chains under <.
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