Discrete Mathematics - Vol 1 | 14. Lecture -14 by Abraham | Learn Smarter
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14. Lecture -14

The chapter explores various mathematical concepts through proof techniques such as induction and strong induction. It demonstrates the relationships between arithmetic and geometric means, binary representations of integers, the definition of a celebrity in a party context, and irrationality proofs for numbers like √2. Additionally, it provides methods for counting diagonals in polygons and emphasizes the importance of clear definitions and logical reasoning in mathematics.

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Sections

  • 14.1

    Discrete Mathematics

    This section introduces fundamental concepts in discrete mathematics, specifically focusing on proof techniques and properties of numbers and arithmetic.

    Learning Practice

  • 14.1.1

    Lecture -14

    In this tutorial, key mathematical proofs are discussed using induction to demonstrate concepts in discrete mathematics, including relationships between arithmetic and geometric means, the representation of integers as sums of powers of two, and the unique identification of a celebrity in a group.

    Learning Practice

  • 14.1.2

    Tutorial 2: Part Ii

    This section covers several mathematical problems and proofs regarding concepts such as induction, means, binary representation of integers, and search for celebrities in a group.

    Learning Practice

  • 14.1.3

    Question Number 8

    This section demonstrates the proof by induction that the arithmetic mean of n arbitrary positive real numbers is greater than or equal to their geometric mean when n is a power of two.

    Learning Practice

  • 14.1.4

    Question Number 9

    This section discusses the proof that every positive integer can be expressed as a sum of distinct powers of two using mathematical induction.

    Learning Practice

  • 14.1.5

    Question Number 10

    This section discusses the concept of identifying a celebrity among guests at a party using the Knows function, utilizing a proof by induction approach.

    Learning Practice

  • 14.1.6

    Question Number 11

    This section presents a proof of the irrationality of √2 using strong induction.

    Learning Practice

  • 14.1.7

    Question Number 12

    In this section, the diagonal formula for polygons is introduced and proved using induction.

    Learning Practice

  • 14.2

    Proof By Induction

    This section explains proof by induction, demonstrating its application in proving universally quantified statements using various examples.

    Learning Practice

  • 14.2.1

    Base Case And Inductive Hypothesis

    This section discusses the principles of proof by induction, focusing on the concepts of base case and inductive hypothesis.

    Learning Practice

  • 14.2.2

    Proceeding With Induction

    This section focuses on using proof by induction to establish specific mathematical statements, including the inequality between arithmetic and geometric means and the unique representation of integers as sums of distinct powers of two.

    Learning Practice

  • 14.2.3

    Conclusion Of Induction

    This section summarizes the principles of mathematical induction, emphasizing the importance of establishing a base case and using inductive steps to prove universal statements.

    Learning Practice

References

ch14.pdf

Class Notes

Memorization

What we have learnt

  • Arithmetic mean is greater ...
  • Every positive integer can ...
  • A celebrity in a social con...

Final Test

Revision Tests