Principle of Mathematical Induction (PMI)

• Definition: The principle of mathematical induction is a method for proving statements about natural numbers.

• Basic Structure:

  • Let S' be a subset of N such that:

    1. 1 ∈ S'

    2. If n ∈ S', then n + 1 ∈ S'.

  • Then S' = N, meaning all natural numbers are included.

Inductive Definitions

• Definition: Inductive definitions define objects for each n ∈ N as follows:

  1. Describe the first object (base case).

  2. Define the (n + 1)th object in terms of the nth object (inductive step).

• Hence, these objects are defined for all n ∈ N.

Examples of Inductive Definitions

• Example 1.2: Terminal Sigma notation

  • Base: [ S(1) = 1 ]

  • Induction: [ S(n) = S(n-1) + n ]

• Example 1.3: Product defined as [ P(n) = n ]

  • Base: [ P(1) = 1 ]

  • Induction: [ P(n + 1) = (n + 1) \cdot P(n) ]

• Example 1.4: Factorial [ n! ]

  • Base: [ 1! = 1 ]

  • Induction: [ (n + 1)! = (n + 1) , n! ]

Proofs by Mathematical Induction

• General Approach:

  1. Base Case: Prove the statement for n = 1.

  2. Inductive Step: Assume that the statement holds for n.

     • Prove that it also holds for n + 1.

• Example: Prove [ orall n ext{ in } N, P(n) ] is true.

  • Basis: Show that [ P(1) ] is true.

  • Inductive Step: Assume [ P(n) ] is true; prove [ P(n + 1) ] is true.

Example Problems

1. Example 1.5: Prove [ (2n - 1) = 1 + 3 + 5 + ... + (2n - 1) = n^2 ]

   • Base Case: For n = 1, [ 1 = 1^2 ] is true.

   • Inductive Step: Assume true for n, prove for n + 1.

2. Example 1.6: Prove [ ext{Product of odds: } orall n: II (2i - 1) = 2n. ]

   • Basis. Step:[ (2i - 1) = 1 ]

   • Induction: Assume true for n, prove for n + 1.

3. Example 1.7: Show [ n^3 + 5n + 6 \text{ is divisible by 3 for all n ∈ N.} ]

   • Basis: For n = 1, [ 1^3 + 5(1) + 6 = 12, ] is true.

   • Induction: Assume true for n, show it for n + 1.

4. Example 1.8: Prove [ 3n > 1 + 2n \forall n ext{ in } N. ]

   • Basis: For n = 1, [ 3 > 3 ] is true.

   • Induction: Assume for n; show for n + 1.

Generalized Mathematical Induction

• Example 1.9: Prove [ 2^n > n^2 \forall n > 4 ] using PMI.

  • Basis: Show true for n=5.

  • Inductive Hypothesis: Assume true for n, prove for n + 1.

Conclusion

• The principle of mathematical induction is a powerful technique for proving a wide range of statements about natural numbers, leveraging both base cases and inductive steps to establish the validity of assertions for all natural numbers.

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