Discrete Mathematics - Vol 2 | 18. Subsequence Existence by Abraham | Learn Smarter
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18. Subsequence Existence

The chapter explores key mathematical principles related to sequences, particularly subsequences that are either strictly increasing or strictly decreasing. It utilizes various mathematical tools, including the pigeonhole principle, to demonstrate the existence of such subsequences in any arbitrary sequence of distinct real numbers. The chapter also delves into problems involving subsets and age groups, highlighting the application of these principles in real-world scenarios.

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Sections

  • 18

    Subsequence Existence

    This section discusses the existence of subsequences within any sequence of distinct real numbers, specifically showing that one can always find a strictly increasing or strictly decreasing subsequence of length n.

    Learning Practice

  • 18.1.1

    Definition Of Increasing And Decreasing Sequences

    This section introduces the concepts of strictly increasing and decreasing sequences, focusing on the existence of subsequences within a given set of distinct real numbers.

    Learning Practice

  • 18.1.2

    Definition Of Subsequences

    The section discusses the concept of subsequences, particularly focusing on the existence of strictly increasing or decreasing subsequences within a given sequence of distinct real numbers.

    Learning Practice

  • 18.1.3

    Universality Of The Statement

    This section discusses the universal property of sequences of distinct real numbers that guarantees the existence of an increasing or decreasing subsequence of length k+1.

    Learning Practice

  • 18.1.4

    Pigeonhole Principle Application

    This section discusses the Pigeonhole Principle and its application in proving the existence of either a strictly increasing or strictly decreasing subsequence in any sequence of distinct real numbers.

    Learning Practice

  • 18.1.5

    Contradiction And Conclusion

    This section discusses the guarantee of subsequences in any sequence of distinct real numbers, emphasizing the existence of either strictly increasing or strictly decreasing subsequences.

    Learning Practice

  • 18.2

    Disjoint Group Sums

    The section analyzes the existence of disjoint groups of people having the same sum in a given range.

    Learning Practice

  • 18.2.1

    Age Group And Pigeonhole Principle Application

    This section explores how to apply the pigeonhole principle to demonstrate the existence of increasing or decreasing subsequences within any selected group of distinct real numbers.

    Learning Practice

  • 18.2.2

    Achieving Disjoint Groups

    This section discusses the establishment of disjoint groups of distinct real numbers ensuring subsequences that are either strictly increasing or decreasing.

    Learning Practice

  • 18.3

    Divisibility In Arbitrary Subsets

    This section demonstrates that any sequence of n+1 distinct real numbers contains a subsequence of length n that is either strictly increasing or strictly decreasing.

    Learning Practice

  • 18.3.1

    Unique Factorization

    This section explores the concept of subsequences in distinct real numbers, proving that any sequence of distinct real numbers contains a subsequence that is either strictly increasing or strictly decreasing.

    Learning Practice

  • 18.3.2

    Pigeonhole Principle Argument

    This section discusses the Pigeonhole Principle and its application in proving the existence of increasing or decreasing subsequences in a sequence of distinct real numbers.

    Learning Practice

  • 18.3.3

    Conclusion On Divisibility

    This section explores the concept of subsequences within distinct real numbers, emphasizing the existence of either strictly increasing or strictly decreasing subsequences.

    Learning Practice

  • 18.4

    Counting Solutions To Equations

    This section explores the existence of subsequences in a series of distinct real numbers, focusing on proving that there will always be a subsequence that is either strictly increasing or strictly decreasing.

    Learning Practice

  • 18.4.1

    Restrictions On Solution Counts

    This section discusses the existence of subsequences in a sequence of distinct real numbers, specifically demonstrating that any sequence of n+1 distinct real numbers contains a subsequence of length n that is either strictly increasing or strictly decreasing.

    Learning Practice

  • 18.4.2

    Applying Restrictions And Solving

    This section explores the application of mathematical principles in proving the existence of specific subsequences within distinct real numbers, highlighting both increasing and decreasing sequences.

    Learning Practice

  • 18.4.3

    Combining Solutions

    This section demonstrates that in any sequence of distinct real numbers, there exists either a strictly increasing or strictly decreasing subsequence of length k+1.

    Learning Practice

References

ch39 - part B.pdf

Class Notes

Memorization

What we have learnt

  • A sequence with distinct re...
  • The pigeonhole principle ca...
  • Mathematical proofs can be ...

Final Test

Revision Tests