Hi, I am using 3.5 version and trying to determine “Axiom Consistency”. Not sure but looks like the order I list the axioms determine the response as to consistent or not. The axioms are consistent however when listed order is changed I get different answers. My assumption is that the order should not matter.

ORDER#1:

Members: A,B,C,D,E,F

Axiom 1: Each committee is a set of three members.

Axiom 2: Each member is on exactly two committees.

Axiom 3: No two members may be together on more than one committee.

Axiom 4: There is at least one committee.

ChatGPT

Let’s examine the given information with the list of members provided:

Members: A, B, C, D, E, F.

Axiom 1: “Each committee is a set of three members.”

Axiom 2: “Each member is on exactly two committees.”

Axiom 3: “No two members may be together on more than one committee.”

Axiom 4: “There is at least one committee.”

Let’s see if these axioms are consistent with each other and whether they can be applied to the list of members:

Axiom 4 states that there is at least one committee, which is valid.

Axiom 2 states that each member is on exactly two committees. Given that there are six members, this implies there must be a total of six committees (6 members * 2 committees/member = 12 committees). However, Axiom 1 contradicts this because it states that “Each committee is a set of three members,” which would imply that each committee has three members, not two.

So, Axiom 2 and Axiom 1 are inconsistent with each other, creating a logical problem in the set of statements.

Axiom 3 states that no two members may be together on more than one committee. This condition is not explicitly violated by the given information but depends on how the committees are formed.

The **inconsistency** arises from Axiom 2 and Axiom 1, which together imply more committees than can be formed with three members each. Therefore, the axioms are not consistent with each other when applied to this specific set of members and committees.

ORDER#2:

Members: A,B,C,D,E,F

Axiom 1: Each committee is a set of three members.

Axiom 2: No two members may be together on more than one committee.

Axiom 3: Each member is on exactly two committees.

Axiom 4: There is at least one committee.

ChatGPT

Let’s use the given axioms to construct a set of committees with members A, B, C, D, E, and F. We’ll follow the axioms step by step.

Axiom 4 states that there is at least one committee. So, let’s start by creating one committee.

Committee 1: {A, B, C}

Now, we move on to Axiom 1, which states that each committee is a set of three members. So, we need to create additional committees while ensuring that no two members are together on more than one committee (Axiom 2) and that each member is on exactly two committees (Axiom 3).

Let’s create Committee 2:

Committee 2: {D, E, F}

Now, we have two committees, and each member is on exactly one committee. We need to add more committees to satisfy Axiom 3. To do this, we’ll move one member from each existing committee to a new committee while adhering to Axiom 2 (ensuring no two members are together on more than one committee).

Let’s create Committee 3:

Committee 3: {A, D, X}

To form Committee 3, we moved A from Committee 1 and D from Committee 2 to create a new committee. We introduced a new member X into Committee 3 to maintain the total number of members. Now, each member is on two committees.

Next, we create Committee 4:

Committee 4: {B, E, Y}

Again, we moved B from Committee 1 and E from Committee 2 to create a new committee, introducing a new member Y.

Finally, we create Committee 5:

Committee 5: {C, F, Z}

This time, we moved C from Committee 1 and F from Committee 2 to create a new committee, introducing a new member Z.

Now, we have five committees, and each member is on exactly two committees. These committees satisfy all the given axioms:

Each committee has three members (Axiom 1).

No two members are together on more than one committee (Axiom 2).

Each member is on exactly two committees (Axiom 3).

There is at least one committee (Axiom 4).

So, the committees are as follows:

Committee 1: {A, B, C}

Committee 2: {D, E, F}

Committee 3: {A, D, X}

Committee 4: {B, E, Y}

Committee 5: {C, F, Z}