Mathematical Induction

Introduction to Mathematical Induction

Mathematical induction is a powerful mathematical proof technique used to prove statements about natural numbers or integers. It is a fundamental tool in discrete mathematics and is commonly used to demonstrate the truth of various mathematical propositions, especially those that involve infinite sets or sequences.

Principle of Mathematical Induction

Mathematical induction is based on the following principle:

Principle of Mathematical Induction: To prove that a statement is true for all positive integers (usually starting from 1 or 0), you need to follow two steps:

  1. Base Case (Initialization): Prove that the statement is true for the first integer (often 1 or 0). This establishes the base case.

  2. Inductive Step (Induction): Prove that if the statement is true for an arbitrary positive integer 'k', then it must also be true for the next integer 'k + 1'. This is called the inductive step.

Steps to Perform Mathematical Induction

  1. Base Case (Initialization):

    • Start by proving that the statement is true for the first integer, often 1 or 0.
    • This is typically a straightforward proof and serves as the foundation for the induction.
  2. Inductive Step (Induction):

    • Assume that the statement is true for some arbitrary positive integer 'k' (the induction hypothesis).
    • Prove that if the statement is true for 'k', it must also be true for 'k + 1'.
    • This often involves manipulating the statement for 'k + 1' using the assumption that it's true for 'k'.
  3. Conclusion:

    • Conclude that, based on the base case and the inductive step, the statement is true for all positive integers.

Example of Mathematical Induction

Let's use an example to illustrate the process of mathematical induction. We want to prove the following statement for all positive integers 'n':

Statement: 1 + 2 + 3 + ... + n = (n * (n + 1)) / 2

Step 1: Base Case

For n = 1, we have:

1 = (1 * (1 + 1)) / 2

This is true, so the base case holds.

Step 2: Inductive Step

Assume the statement is true for some positive integer 'k'. That is,

1 + 2 + 3 + ... + k = (k * (k + 1)) / 2

Now, we need to prove it for 'k + 1'. Add (k + 1) to both sides of the equation:

1 + 2 + 3 + ... + k + (k + 1) = (k * (k + 1)) / 2 + (k + 1)

Using the induction hypothesis:

(k * (k + 1)) / 2 + (k + 1) = (k * (k + 1) + 2(k + 1)) / 2

Simplify the right side:

(k * (k + 1) + 2(k + 1)) / 2 = ((k + 1) * (k + 2)) / 2

So, the statement is also true for 'k + 1'.

Step 3: Conclusion

By mathematical induction, the statement is true for all positive integers.

Summary

Mathematical induction is a powerful method for proving statements about natural numbers or integers. It consists of a base case, where you establish the statement's truth for the first integer, and an inductive step, where you prove that if the statement holds for an arbitrary integer 'k', it also holds for 'k + 1'. This process allows you to extend the proof to all positive integers. It is a valuable tool in mathematics and is commonly used in various mathematical contexts.