Book cover for Discrete Mathematics with Applications

Discrete Mathematics with Applications

Thomas Koshy

ISBN #124211801

1st Edition

3,631 Questions

Group icon
38,965 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter delves into the fundamental principles of number theory and algorithms, emphasizing the well-ordering principle, the division algorithm, and mathematical induction. Through clear definitions and problem-solving examples, the text demonstrates how these concepts underpin proofs and offer versatile tools for tackling diverse mathematical problems. Additionally, by integrating the pigeonhole principle, the chapter highlights the interconnectivity of combinatorial thinking with traditional number theory, providing a solid basis for further exploration in algorithms.

Learning Objectives

1

Understand and articulate the well-ordering principle and its significance in proofs and number theory.

2

Explain the division algorithm and identify its unique components: the quotient and the remainder.

3

Apply the principle of mathematical induction to prove propositions and solve problems.

4

Utilize fundamental divisibility properties and the pigeonhole principle to explore and solve intriguing numerical problems.

Key Concepts

CONCEPT

DEFINITION

Well-Ordering Principle

The assertion that every nonempty set of positive integers has a least element. This principle is often used as a basis for constructing proofs.

Division Algorithm

A theorem stating that for any integer a and any positive integer b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r and 0 ≤ r < b.

Mathematical Induction

A proof technique that establishes the truth of an infinite sequence of statements by proving a base case and an inductive step, thereby showing that if one case holds, the next does as well.

Divisibility Properties

Rules and attributes that dictate how integers divide one another, often used to derive further number theoretic conclusions.

Pigeonhole Principle

A counting principle that states if more objects are distributed into fewer containers, then at least one container must hold more than one object. It is widely used in combinatorial arguments.

Example Problems

Example 1

Is the set of positive odd integers well-ordered?

Example 2

Is the set of positive even integers well-ordered?

Example 3

In Exercises $3-6,$ find the quotient and the remainder when the first integer is divided by the second. $$ 137,11 $$

Example 4

In Exercises $3-6,$ find the quotient and the remainder when the first integer is divided by the second. $$ 15,23 $$

Example 5

In Exercises $3-6,$ find the quotient and the remainder when the first integer is divided by the second. $$ -43,16 $$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Prove that the sum of the first n positive integers is given by 1 + 2 + ... + n = n(n + 1)/2 using mathematical induction.

STEP-BY-STEP ANSWER:

Step 1: Base Case - Verify the formula for n = 1. Since 1 = 1(1 + 1)/2 = 1, the base case holds.
Step 2: Inductive Hypothesis - Assume the formula is true for some positive integer k, i.e., 1 + 2 + ... + k = k(k + 1)/2.
Step 3: Inductive Step - Consider the statement for n = k + 1. Start with the left-hand side: 1 + 2 + ... + k + (k + 1). By the inductive hypothesis, this equals k(k + 1)/2 + (k + 1).
Step 4: Factor the Expression - Factor out (k + 1): k(k + 1)/2 + (k + 1) = (k + 1)(k/2 + 1) = (k + 1)( (k + 2)/2 ).
Step 5: Conclude - Simplify to obtain (k + 1)(k + 2)/2, which is exactly the formula for n = k + 1. Thus the formula is valid for k + 1.
Final Answer: By the principle of mathematical induction, the formula 1 + 2 + ... + n = n(n + 1)/2 holds for all positive integers n.

Mathematical Induction

QUESTION

Given an integer a = 37 and a positive integer b = 5, use the division algorithm to find the unique quotient and remainder.

STEP-BY-STEP ANSWER:

Step 1: Identify a and b. Here a = 37 and b = 5.
Step 2: Determine the quotient q by finding the greatest integer such that 5q ≤ 37. In this case, q = 7 because 5 × 7 = 35 and 5 × 8 = 40 > 37.
Step 3: Compute the remainder r: r = 37 - 5q = 37 - 35 = 2.
Step 4: Verify the condition 0 ≤ r < 5, which holds since 0 ≤ 2 < 5.
Final Answer: The quotient is 7 and the remainder is 2, so 37 = 5 × 7 + 2.

Division Algorithm

Scroll left
Scroll right

Common Mistakes

  • Failing to properly establish the base case in an induction proof, leading to incorrect or incomplete arguments.
  • Misidentifying the quotient and remainder in the division algorithm by neglecting the uniqueness condition.
  • Overlooking or misunderstanding the specific conditions required for the application of the pigeonhole principle.
  • Assuming that induction proves the truth of a statement for all cases without adequately verifying the inductive step.