Hawraa Abbas Almurieb

Consistency

Independence

Completeness

Tennis Game Undefined Terms: game(line), player(point) (a)For each two players, there is a game for them. (b)There are at least 3 players. (c) Each game consists of two players only.

Players: Ali, Ahmed, Abbas, Zaid.

Games:

G1: Ali, Ahmed

G2: Ali, Abbas

G3: Ali, Zaid

G4: Ahmed, Abbas

G5: Ahmed, Zaid

G6: Abbas, Zaid

Affine planes are independent iff

1. Ax1 is independent:

(~Ax1,Ax2,Ax3,Ax4) is consistent

2. Ax2 is independent:

(Ax1, ~Ax2,Ax3,Ax4) is consistent

3. Ax3 is independent:

(Ax1,Ax2, ~Ax3,Ax4) is consistent

4. Ax4 is independent:

(Ax1,Ax2,Ax3, ~Ax4) is consistent

FIFA World Cup Model

Undefined Terms: national team (point),

group (line)

(a) There are 32 national teams and 8

groups.

(b) Every group contains four national teams.

(c) Every national team is on exactly one

group.

National Teams(Points):

Groups(Lines): A, B , C , D , E , F , G , H.

National team (point)

Groups (lines)

(Ax1, ~Ax2,Ax3,Ax4) is consistent

Number and sets Model:

Undefined terms: number (point), set(line)

(a) For each two numbers, there is a set

containing them.

(b) Not all numbers in the same set.

(c) For a number a and a set A, a∉A , there is

a set B s.t. a∊B.

Numbers(point): 1,2,3,4,5.

Sets(lines): {1,2,3}, {1,4}, {3,4,5}

1

5

3 2

4

(Ax1, Ax2, ~Ax3,Ax4) is consistent

3-point Geometry:

Undefined terms: point, line

Given any two distinct points, there is a

unique line passing through both of them.

There are three points.

There is one line.

(Ax1, Ax2,Ax3, ~Ax4) is consistent

Undefined terms: members(points) and

parties(lines)

Defined term: People Representative is a

member that is exists in every party.

(a)There are exactly five members.

(b)People representative is a member on each

party.

Members: Omer, Hussein, Sura, Aws, Muna.

People representative: Aws

Parties:

P1: Aws, Sura, Hussien.

P2: Aws, Omer, Muna.

Model1: Numbers{1,2,3,4,5},

Lines{12,13,14,15,23,24,25,34,35,45}

Model2: Letters {A,B,C,D,E }, Lines {AB,

,BC, AC,BE ,AE,BD,CD,CE, AD,DE}

2

4 3

5

1

1→A

2→B

3→C

4→D

5→E

2

4 3

5

1

B

D C

E

A

Show that the projective geometry is :

1. consistent

2. independent

3. complete