math logic and axiom

Independence

Name:

Create an independence model for the following axioms.

1. Axiom System II

The Primitive Terms: element, set, and in

Axiom 1: There exists at least one set.

Axiom 2: For each element, there exists exactly one other element such that no set contains both.

Axiom 3: Each set contains exactly two elements.

Axiom 4: Each element is contained in exactly four sets.

Solution: Let elements by points and sets be lines.

Axiom 1: The empty graph

Axiom 2:

Axiom 3:

Axiom 4:

2. Axiom System III The Primitive Terms: point, line, and on

Consider the following definitions:

Definition 1. Two lines are parallel if they are not on any common point.

Definition 2. Three or more points are called noncollinear if they are not all on one common line.

Consider the axioms:

Axiom 1: Every line is on at least one point.

Axiom 2: There exist at least two points.

Axiom 3: Any two distinct points are on exactly one common line.

Axiom 4: For each line there is at least one point not on it.

Axiom 5: If ` is a line and P is a point not on `, then there exists exactly one line on P parallel to `.

Solution:

Axiom 1:

By Axiom 2, there exists at least two points, say p1 and p2. By Axiom 3, there exists a common line, l1 that p1 and p2 are on. By Axiom 4, there is at least one point p3 not on l1. By Axiom 5, p3 much be on a line l2 that is parallel to l1. Now suppose there is a line l3 that does not have any points on it.

Axiom 2:

The empty graph

Axiom 3:

Axiom 4:

Axiom 5:

3. Axiom System IV

The Primitive Terms: player, team, and recruit

Axiom 1: There exists at least one team.

Axiom 2: Each team recruits exactly two players.

Axiom 3: For each team, there is exactly one other team such that no player is recruited by both.

Axiom 4: Each player is recruited by exactly two teams.

Solution: Let players be points and teams be lines

Axiom 1: The empty graph

Axiom 2:

Axiom 3:

Axiom 4:

4. Axiom System V

The Primitive Terms: citizen, superhero, protect

Axiom 1: There exists at least one superhero.

Axiom 2: Each superhero protects exactly three citizens.

Axiom 3: Each pair of superheroes protects exactly two citizens in common.

Axiom 4: Each pair of citizens is protected by exactly two common superheroes.

Solution: Let citizens be points and superheroes be lines

Axiom 1: The empty graph

Axiom 2:

Axiom 3:

Axiom 4: