Lecture 012

Prime Number: A natural number n is called prime iff n > 1 and its only positive divisors are 1 and n.

The Fundamental Theorem of Arithmetic

Claim: (\forall n \in \mathbb{N}) with n > 1 n is either prime or a product of primes.

Proof: We proceed by strong induction on n\in \mathbb{Z} \land n \geq 2. Define P(n) to be the variable proposition "n either prime or a product of primes"

The Game of Chomp

The Game of Chomp
Claim: If Player I begins the game by choosing square (1, n) the top right corner, then Player 1 can always win the 2 x n game of Chomp. Proof: by strong induction. Let P(n) be the proposition.

This is a strong induction because it require 2 x 1 is true to 2 x k.

Well Ordering Property

Definition: A set X is called well-ordered iff every nonempty subset has a least element.

Claim: \mathbb{N} is well-ordered Proof: by SPMI for all n\in\mathbb{N}. Let P(n) be variable proposition "every subsets of N containing n has a least element" (read: assume the subset containing n has the property, prove subset containing all number has property)

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