Partial Solutions for Axiomatic Systems Exercise 1.3.
There are three kinds of mathematicians: those that can count and those that cannot.
—Author Unknown

Exercise 1.3. Consider the following axiom set.

P1. Every herd is a collection of cows.
P2. There exist at least two cows.
P3. For any two cows, there exists one and only one herd containing both cows.
P4. For any herd, there exists a cow not in the herd.
P5. For any herd and any cow not in the herd, there exists one and only one other herd containing the cow and not containing any cow that is in the given herd.

1. What are the primitive terms in this axiom set?
   Elements: herd, cow.  Relations: collection/contain/in.
    

2. Deduce the following theorems:
       T1. Every cow is contained in at least two herds.
   Proof. Let A be a cow. By P2, there is a second cow B that is not cow A. By P3, there is one and only one herd h that contains both cows A and B. By P4, there is a cow C that is not in herd h and is distinct from cows A and B. Thus, by P3, there is one and only one herd k that contains cows A and C. Since C is not in herd h, herds h and k are two distinct herds that contain cow A. Since cow A was arbitrarily chosen, every cow is contained in at least two herds.//
   
   Alternate Proof. Let A be a cow. By P2, there is a second cow B that is not cow A. By P3, there is one and only one herd h that contains both cows A and B. By P4, there is a cow C that is not in herd h and is distinct from cows A and B. Thus, by P3, there is one and only one herd k that contains cows B and C. Hence, since cow C is not in herd h and h is the only herd that contains cows A and B, cow A is not in herd k. Therefore, by P5, there is one and only one herd m that contains cow A and does not contain either B or C. Hence, herds m and h are two herds that contain cow A. Since cow A was arbitrarily chosen, every cow is contained in at least two herds.//
   
       T2. There exist at least four distinct cows.
   Proof. By P2, there exist two distinct cows A and B. By P3, there is one and only one herd h that contains both cows A and B. By P4, there is a cow C that is not in herd h1 and is distinct from cows A and B. Thus, by P3, there is one and only one herd h2 that contains cows A and C and one and only one herd h3 that contains cows B and C. By P5, there exists herd k1 containing cow A and neither cow B nor C and there exists herd k2 containing cow B and neither cow A nor C. Thus, by P5, either herd h3 or k2 contain a cow D in common with herd k1. Since cow A is contained in neither herd h3 nor herd k2 and cows B and C are not in herd k1, cow D is a distinct fourth cow. Therefore, there exist at least four cows.//
   
       T3. There exist at least six distinct herds.

3. Find two isomorphic models.

4. Demonstrate the independence of the axioms.
   Hints:
   For P1, consider putting a sheep in one of the herds.
   For P2, consider a model with no cows and no herds. Or, consider a model with one cow and no herds.
   For P3, consider a model with two cows and no herds. Or, consider a diagram model of four dots forming a square, i.e. four cows and four herds in a square.
   For P4, consider a model with two cows and one herd.
   For P5, consider a model with three cows and three herds.
   Make sure you justify that your models satisfy or do not satisfy the axioms. Also, note that there are many other models that are possible.