I hope I posted this in the right part of the forum...
Consider the following axiom set, in which x's, and y's, and "on" are the undefined terms:
Axiom 1. There exist exactly 5 x's.
Axiom 2. Any two distinct x's have exactly one y on both of them.
Axiom 3. Each y is on exactly two x's.
1. Prove how many y's are in the system.
2. Prove that any two y's have at most one x on both.
3. Prove that not all x's are on the same y.
4. Prove that there exist exactly 4 y's on each x.
5. Prove that for any y1 and any x1 not on that y1 there exist exactly two other distinct y's on x1 that do not contain any of the x's on y1.