MATHEMATICAL INDUCTION - DISCRETE MATHEMATICS

Introduction

This tutorial covers the concept of mathematical induction, a fundamental proof technique in discrete mathematics. It is particularly useful for proving statements about integers or sequences. By following this guide, you will learn the steps involved in applying mathematical induction and see examples to solidify your understanding.

Step 1: Understand the Principle of Mathematical Induction

Mathematical induction consists of two main steps:

1. Base Case: Prove the statement for the initial value, usually n = 1.
2. Inductive Step: Assume the statement is true for some integer k (the inductive hypothesis), and then prove that it must also be true for k + 1.

Practical Advice

• Ensure that the base case is correctly established; this is crucial for induction to hold.
• The inductive step should clearly show how the truth of the statement for k implies its truth for k + 1.

Step 2: Prove a Simple Statement Using Induction

Let's prove the statement: "The sum of the first n natural numbers is given by the formula:

[ S(n) = \frac{n(n + 1)}{2} ]

Base Case

• For n = 1:
  • Left side: S(1) = 1
  • Right side: (\frac{1(1 + 1)}{2} = 1)

Both sides are equal; thus, the base case holds.

Inductive Step

1. Assume true for n = k: [ S(k) = \frac{k(k + 1)}{2} ]

2. Show it holds for n = k + 1: [ S(k + 1) = S(k) + (k + 1) ] Substituting the inductive hypothesis: [ S(k + 1) = \frac{k(k + 1)}{2} + (k + 1) ] Combine the terms: [ S(k + 1) = \frac{k(k + 1) + 2(k + 1)}{2} = \frac{(k + 1)(k + 2)}{2} ] This matches the formula for n = k + 1.

Conclusion of the Proof

Since both the base case and inductive step are proven, the statement holds for all natural numbers n.

Step 3: Explore Common Pitfalls

• Neglecting the Base Case: Always verify your base case, as it's critical for the proof.
• Assuming the Inductive Hypothesis is True: It’s essential to clearly show how it leads to the case for k + 1.
• Overcomplicating the Inductive Step: Keep your algebra simple and clear to avoid mistakes.

Conclusion

Mathematical induction is a powerful tool for proving statements related to natural numbers. By mastering the base case and inductive step, you'll be equipped to tackle a variety of proofs in discrete mathematics. For further practice, explore more examples of mathematical induction, and consider applying it to different mathematical statements or sequences.