I hope I posted this in the right part of the forum...

Consider the following axiom set, in whichx's, andy's, and "on" are the undefined terms:

Axiom 1.There exist exactly 5x's.

Axiom 2.Any two distinctx's have exactly oneyon both of them.

Axiom 3.Eachyis on exactly twox's.

1. Prove how manyy's are in the system.

2. Prove that any twoy's have at most onexon both.

3. Prove that not allx's are on the samey.

4. Prove that there exist exactly 4y's on eachx.

5. Prove that for anyy1 and anyx1 not on thaty1 there exist exactly two other distincty's onx1 that do not contain any of thex's ony1.