1. ## axiomatic systems

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

Undefined terms: Fe's, Fo's, and the relation "belongs to."

Axiom 1. There exist exactly three distinct Fe's in this system.
Axiom 2. Any two distinct Fe's belong to exactly one Fo.
Axiom 3. Not all Fe's belong to the same Fo.
Axiom 4. Any two distinct Fo's contain at least one Fe that belongs to both.

Prove: There exists a set of two Fo's that contains all the Fe's of the system.

2. By A1 we know there are only 3 distinct Fe's, let call them them Fe1, Fe2, Fe3. By A2, we know there exists a Fo, we'll call it Fo1 such that Fe1, Fe2 belong to Fo1. Similarly we can find another Fo such that Fe2, Fe3 belong to it and by A3, it cannot be the same as Fo1, so let's call it Fo2. So we now know Fe1, Fe2 belong to Fo1 and Fe2, Fe3 belong to Fo2, and we notice that Fe2 belongs to both Fo1 and Fo2, so A4 is not broken; and we have found a set {Fo1, Fo2} that contains all of the Fe's of the system.

I see that you posted several questions. Now that you have 1 to go on, try them yourself, they all should follow similar logic, if you're still stuck on something, post what you have done so we can help with the next step and not depriving you of good practice.