Clarification about a given axiom system.

Question

I am now currently studying Combinatorics of Finite Geometries. One problem asks if the given axiom system below is consistent or inconsistent.

1. There are five points and six lines.
2. Each point is in at most two lines.
3. Each line contains two points.

Is the given axiom system consistent with a sample structure given below?

My answer is no since axiom 2 will be violated. In particular there are points that are contained in six lines.

My questions are: (1) Am I correct? (2) If I am correct is there a possible structure that satisfies the given axiom system?

Thanks for the help.

Answer 1

(1) You are correct. I hope your teacher isn't the one who thought the system in the picture satisfies the axioms.

(2) No, the given axiom system is not satisfiable. How many pairs $(p,L)$ are there, consisting of a point $p$ which is on a line $L$? By axioms 1 and 2, there are at most $10$ such pairs; but by axioms 1 and 3, there must be $12$ of them. The axioms are inconsistent.