Tournament System Axiom 1 Every game consists of two players

Tournament System Axiom 1: Every game consists of two players. Axiom 2: For every distinct pair of players, p and p, there is exactly one game. Axiom 3: There are exactly M players. a. What are the undefined terms? b. How many games will there be? c. Is the system consistent? (Show a model for M = 5. Explain what happens each time M increases by 1. 3 points.) d. Is the system independent? (Your answer requires proof. 3 points.)

Solution

An axiomatic system is a list of undefined terms together with a list of statements (called “axioms”) that
are presupposed to be “true.” A theorem is any statement that can be proven using logical deduction
from the axioms.

a.Undefined terms: Games and players

b.Number of games: M-1 (as there are 1 less game than the number of player if they play each other once)

c.Independence: An axiom is called independent if it cannot be proven from the other axioms. In other words, the axiom “needs” to be there, since you can’t get it as a theorem if you leave it out. How do you prove that
something can’t be proved? This relates to the area of mathematics known as logic.

d.Consistency:- If there is a model for an axiomatic system, then the system is called consistent. Otherwise, the system is inconsistent. In order to prove that a system is consistent, all we need to do is come up with a model:
a definition of the undefined terms where the axioms are all true. In order to prove that a system is
inconsistent, we have to somehow prove that no such model exists

For M=5 there are exactly 4 games should be played. Hence there is consistency in the system.Since this is a definition of the undefined terms where the axioms are all true