Basis for Recursion

Induction

"Infinite Ladder"

Assume that there's a ladder with infinite rungs. Can we reach every step of the ladder (mathematically)?

"Infinite Ladder"

Let P(n) be "Reach rung n of the ladder".

 

Prove two things.

 

* We can reach the first rung of the ladder.

* Given a rung k, assume P(k). Prove P(k + 1)

Mathematical Induction

We will prove a proposition P on all positive integers n. To do so, we need to prove two things.

 

Basis Step: Prove that P(1) is true.

Inductive Step: Show that the conditional P(k) → P(k + 1) is true for all integers k.

Mathematical Induction

As a single rule of inference.

 

(P(1) ∧ ∀k (P(k) → P(k + 1)) = ∀n P(n)

Strong Induction

Strong vs. Regular

Recall the definition of induction.

 

Basis Step: Prove P(b)

Inductive Step: Prove P(k) → P(k + 1)

Strong vs. Regular

Strong induction is very similar.

 

Basis Step: Prove P(b)

 

Inductive Step:

Prove [P(b) ^ P(b + 1) ^ P(b + 2) ... P(k)] -> P(k + 1)

What's the Diff?

Sometimes you need more than P(k).

 

Consider: Prove that any integer > 1 can be written as the product of primes.