Coming up with Theorems and proving them

[Zoey93]

Axioms 1 and 2 were given and axioms 3 and 4 I came up with:

Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: There exist at least one game.
Axiom 4: Each team plays at most 10 games.

I need help with coming up with theorems for these axioms and proving them.

 

[Ackbach]

Hmm. With existence axioms that weak, you might not have too many theorems you can prove. One model for your system is the following:

*   *
|
*   *

Here the * is a team, and the | is a game. This satisfies all four axioms. I suppose one theorem you could come up with is that there are at least two teams that play a game. It's not a very advanced theorem, but like I said, you don't have terribly strong existence axioms. I don't think you're guaranteed much structure beyond this model.

So let's take this theorem: how could you prove it?

 

[Zoey93]

Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: There exist at least one game.
Axiom 4: Each team plays at most 10 games.

Theorem 1: There are at least two teams that play a game.

Proof: According to Axiom 1, each game is played by two distinct teams, which we will call team A and team B. By Axiom 3, there exist at least one game.

I need help finishing up the proof.

 

[Ackbach]

Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: There exist at least one game.
Axiom 4: Each team plays at most 10 games.

Theorem 1: There are at least two teams that play a game.

Proof: According to Axiom 1, each game is played by two distinct teams, which we will call team A and team B. By Axiom 3, there exist at least one game.

I need help finishing up the proof.

I think it would make more sense to start with your second line, so that it runs like this:

By Axiom 3, there exists at least one game, call it 'g'. By Axiom 1, 'g' must be played by two distinct teams, call them 'A' and 'B'. These are the two teams in the theorem, hence we are done.

You could strengthen your theorem just a bit by saying "There are at least two distinct teams that play a game." The proof would be no more complicated.

 

[Zoey93]

Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: There exist at least one game.
Axiom 4: Each team plays at most 10 games. I came up with another theorem because my professor wanted us to come up with two theorems. So, far this is what I got:

Theorem 2: If there are exactly 4 distinct teams, then there are at most 20 games.

Proof: By hypothesis we have exactly 4 distinct teams (which is allowed by Axiom 2), and I will call them A, B, C, and D. By Axiom 4, each team plays at most 10 games, but Axiom 3 requires there to exist at least one game between the teams.

But I need help to finish it up.

 

[Ackbach]

Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: There exist at least one game.
Axiom 4: Each team plays at most 10 games. I came up with another theorem because my professor wanted us to come up with two theorems. So, far this is what I got:

Theorem 2: If there are exactly 4 distinct teams, then there are at most 20 games.

Proof: By hypothesis we have exactly 4 distinct teams (which is allowed by Axiom 2), and I will call them A, B, C, and D. By Axiom 4, each team plays at most 10 games, but Axiom 3 requires there to exist at least one game between the teams.

But I need help to finish it up.

Axiom 3 isn't going to help you here, because you're already thinking about the other direction - the maximum number of games allowed by your axioms. The minimum number of games is 1. To get the maximum number of games, you're going to need to think about graph theory, essentially. Try to draw representations of various scenarios. What if there are the maximum 10 games between A and B? Could A or B then have any games with C or D? Or what if it's more evenly distributed? Suppose there are 3 games between A and B, 3 between A and D, and 4 between A and C?

I feel like there's an elegant way to prove this theorem, but it's eluding me at the moment. Something along the lines of this: there are four teams. Each team can play a maximum of ten games. But each of those games must involve exactly one other team. Therefore, although, from each team's perspective, there are 40 games maximum, since each of these games must involve another team, and "use up" one of their games, there can only be a maximum of 20 games. This is not rigorous, but it might give you the idea of the proof.

 

[Zoey93]

Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: There exist at least one game.
Axiom 4: Each team plays at most 10 games. Theorem 2: If there are exactly 4 distinct teams, then there are at most 20 games.

Proof: By hypothesis we have exactly 4 distinct teams (which is allowed by Axiom 2), and I will call them A, B, C, and D. By Axiom 4, each team plays at most 10 games. Each team can play a maximum of ten games. But each of those games must involve exactly one other team. Therefore, although, from each team’s perspective, there are 40 games maximum, since each of these games must involve another team, and “use up” one of their games, there can only be a maximum of 20 games.

So, I turned this into my professor and this is what he said: For your proof of Theorem 2, you seem to be lacking a reference to an important axiom. In particular, when you say this “But each of those games must involve exactly one other team.” What is the reason for that statement?

I was thinking maybe I should also use Axiom 1 to prove this theorem??

 

[Ackbach]

Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: There exist at least one game.
Axiom 4: Each team plays at most 10 games. Theorem 2: If there are exactly 4 distinct teams, then there are at most 20 games.

Proof: By hypothesis we have exactly 4 distinct teams (which is allowed by Axiom 2), and I will call them A, B, C, and D. By Axiom 4, each team plays at most 10 games. Each team can play a maximum of ten games. But each of those games must involve exactly one other team. Therefore, although, from each team’s perspective, there are 40 games maximum, since each of these games must involve another team, and “use up” one of their games, there can only be a maximum of 20 games.

So, I turned this into my professor and this is what he said: For your proof of Theorem 2, you seem to be lacking a reference to an important axiom. In particular, when you say this “But each of those games must involve exactly one other team.” What is the reason for that statement?

I was thinking maybe I should also use Axiom 1 to prove this theorem??

Yep! It's a more-or-less direct implication.