Partial Solutions for Axiomatic Systems Exercise 1.2.
You cannot have the success without the failures.
—H. G. Hasler (1914–1987)

Exercise 1.2. Consider the following axiom set.

A1.   Every hive is a collection of bees.
A2.   Any two distinct hives have one and only one bee in common.
A3.   Every bee belongs to two and only two hives.
A4.   There are exactly four hives.

1. What are the undefined terms in this axiom set?

2. Deduce the following theorems:
   T1. There are exactly six bees.
   Proof. By A4, there are exactly four hives, call them H₁, H₂, H₃, H₄. Then by A2, there are bees b₁₂, b₁₃, b₁₄, b₂₃, b₂₄, b₃₄. Let b be any bee. By A3, b must be in two and only two hives, say H_i and H_j for i,j ε {1, 2, 3, 4},    i ≠ j. Then b = b_ij, i,j ε {1, 2, 3, 4}, using combinatorics ₄C₂ = 6  . Hence there are exactly six bees.//
   
   T2. There are exactly three bees in each hive.
   T3. For each bee there is exactly one other bee not in the same hive with it.

3. Find two isomorphic models.

4. Demonstrate the independence of the axioms.
   Here are some sample models, without justification, that demonstrate the independence of the axioms. Can you justify each model?
   A model where A2, A3 and A4 are true, but A1 is not true.
           Hives:  1, 2, 3, 4 each contain the corresponding bees and a bird is contained in each hive.
           Bees:  {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}
   A model where A1, A3 and A4 are true, but A2 is not true.
           Hives:  1, 2, 3, 4
           Bees:   {1, 2}, {1, 3}, {2, 4},{3, 4}
   A model where A1, A2 and A4 are true, but A3 is not true.
           Hives:  1, 2, 3, 4
           Bees:   {1, 2, 3}, {1, 4}, {2, 4}, {3, 4}
   A model where A1, A2 and A3 are true, but A4 is not true.
           Hives:  1, 2
           Bees:   {1, 2}