Well ordering principle

WOP

WOP Description:

• Every non-empty set of non-negative integers has a smallest element

Factoring Primes

Claim 2:

• Any positive integers m and n, the fraction $\frac{m}{n}$ can be written in lowest terms
  • as they are positive int, we can simplify the fraction by dividing with $gcd(m,n)$ → lowest terms

Theorem 3:

• Every positive integer greater than 1 can be factored as a product of primes
  • 3 = 3* 1
  • 9 = 3 * 3
  • 10 = 2 * 5

Round-robin tournament

Definition

• a competition where each player meets every other players, contrast with elimination (eliminated after certain losses)
• $p_1$ beats $p_2$, … , $p_n-1$ beats $p_n$, $p_n$ beats $p_1$
• see the directed graph representation here,edges go from winner to loser

Claim 4 ❗️❗️❗️

• If there is cycle length m ($m\ge 3$) among players in round-robin tournament, there must be a cycle of three of these players
  • a cycle is a sequence of players such that $p_1$ beats $p_2$, … , $p_n$ beats $p_1$

Well Ordered Sets

Theorem 5:

• For any nonnegative int n, the set of integers $\ge -n$ is well ordered
  • consider this is the set S
  • increase each element in the set by n → not change relative order → apply the WOP → there is a smallest element in the set S (denote as m)
  • we get $m-n$ is the actual smallest element of S

Definition 6

• lower bound of set S is $b\le s$, for every $s\in S$, $S$ is set of real numbers) → b is the smallest number in S
• upper bound of set S is $b\ge s$, for every $s\in S$, $S$ is set of real numbers) → b is the largest number in S

Corollary 7

• any set of integers with a lower bound is well ordered