12 Interpret a tree as a committee and a row as a person Use

12. Interpret a \"tree\" as a committee and a row\" as a person. Use the theory developed for the axiom system A-E to assign 6 people Arias, Braun, Chang, Desai, Enoki, and Flynn to 4 committees. 13. Define a triangle to be a set of three lines, each pair of which shares exactly one point, called a vertex. How many triangles are there in the six-point model? 14. Two lines are parallel if they have no points in common. Consider the following parallel postulate: given a line c and a point P not on C, there is a unique line M that contains P and is parallel to C. Do the 4-point geometry and 6-point geometries satisfy this postulate?

Solution

Now as per given assumption let us first right the axioms

Atleast one commitee

Person belongs to exactly two committee

Comittee belongs to atleast one person

Any 2 committe have exactly 1 person in common

For every person there is exactly one other person for which there is no committe in common.

And as per above Axioms we can assign as belwo

1 Arias , Enoki , Chang
2 Braun , Flynn, Enoki
3 Chang , Arias, Desai
4 Desai , Braun , Flynn

Here lets say committe are from 1 to 4 so as per axioms above would be the setting.

Arais - Belongs to two committe and uncommon perosn is Braun
Braun - Belongs to two committe and uncommon perosn is Arias
Chang - Belongs to two committe and uncommon perosn is Braun

Desai - Belongs to two committe and uncommon perosn is Enoki

Enoki - Belongs to two committe and uncommon perosn is Desai

Flynn - Belongs to two committe and uncommon perosn is Chang