Discrete Mathematics - Vol 2 | 17. Module No#08 by Abraham | Learn Smarter
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17. Module No#08

This chapter delves into the principles of discrete mathematics, specifically focusing on the application of the pigeonhole principle to prove the existence of certain properties among sets of integers or points in a plane. The fundamental aim is to showcase how pigeonhole logic can help derive relationships, affirm conditions, and establish the validity of mathematical statements across different scenarios.

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Sections

  • 17.1

    Discrete Mathematics

    This section covers key concepts in discrete mathematics using the pigeonhole principle to prove the existence of certain properties among integers and points in a coordinate system.

    Learning Practice

  • 17.2

    Module No#08

    This section discusses the application of the pigeonhole principle in proving mathematical statements related to midpoint calculations of integer coordinates and the existence of integer pairs whose sum equals a specific number.

    Learning Practice

  • 17.3

    Lecture No#38

    This lecture explores the application of the pigeonhole principle in proving mathematical statements related to integer coordinates and specific sums.

    Learning Practice

  • 17.4

    Tutorial 6: Part Ii

    This section discusses the application of the pigeonhole principle in proving the existence of certain mathematical properties involving distinct points in two-dimensional planes and integers.

    Learning Practice

  • 17.5

    Question 8

    This section discusses the application of the pigeonhole principle to prove the existence of a pair of distinct points in a two-dimensional plane with integer coordinates whose midpoint also has integer coordinates.

    Learning Practice

  • 17.5.1

    Midpoint Of The Line Joining Points

    This section discusses the midpoint of the line segment defined by two arbitrary distinct points in a 2-dimensional integer plane and demonstrates the consistency of their midpoints using the pigeonhole principle.

    Learning Practice

  • 17.5.2

    Application Of Pigeonhole Principle

    The section discusses the application of the Pigeonhole Principle in demonstrating certain properties of points in a 2D plane.

    Learning Practice

  • 17.6

    Question 9

    In this section, the pigeonhole principle is applied to show that among any 5 integers chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}, there always exists at least one pair of integers whose sum is 9.

    Learning Practice

  • 17.6.1

    Identifying Pigeons And Holes

    This section illustrates the application of the pigeonhole principle through examples involving coordinate points and integers.

    Learning Practice

  • 17.6.2

    Mapping To Ordered Pairs

    This section explores the concept of mapping arbitrary distinct points in a 2D plane to ordered pairs, demonstrating the existence of midpoints with integer coordinates using the Pigeonhole Principle.

    Learning Practice

  • 17.7

    Question 10

    In this section, the pigeonhole principle is used to demonstrate the existence of multiples of an integer that consist solely of the digits 0 and 1 in decimal form.

    Learning Practice

  • 17.7.1

    Universally Quantified Statement

    This section discusses the universally quantified statements and utilizes the pigeonhole principle to prove certain mathematical assertions about integers and their properties.

    Learning Practice

  • 17.7.2

    Proof Using Pigeonhole Principle

    This section introduces the Pigeonhole Principle and its application in proving the existence of certain pairs within sets.

    Learning Practice

References

ch39 - part A.pdf

Class Notes

Memorization

What we have learnt

  • The pigeonhole principle ca...
  • Irrespective of the arbitra...
  • Mathematical proofs can dem...

Final Test

Revision Tests