Indirect Proofs

Date: 09/21/97 at 00:30:35
From: Ashley
Subject: Geometry - Indirect Proofs

Dear Dr. Math,

Can you please explain to me what indirect proofs are, and how to 
solve this problem, or at least give me a hint?

If Clark is a mathemagician, then Lois is his assistant. If Clark is a 
mathemagician, then everyone at the theater has a good time.  If 
everyone in the theater has a good time, then Lois is not his 
assistant. Therefore, Clark is not a mathemagician.  

I really need help with this one. Thanks!

Date: 09/21/97 at 16:35:51
From: Doctor Jodi
Subject: Re: Geometry - Indirect Proofs

Hi there!

I'm not quite sure what you mean by indirect proofs, but I think I can
help you with your puzzle. First of all, let's look at the sentences 
that you have:

  (1) If Clark is a mathemagician, then Lois is his assistant. 

  (2) If Clark is a mathemagician, then everyone at the theater 
      has a good time.  

  (3) If everyone in the theater has a good time, then Lois is not 
      his assistant.  

  (4) Therefore, Clark is not a mathemagician.  

Let's focus on the first three sentences for now. Sentences (1), (2), 
and (3) are conditional, if-then sentences. We want to find out 
whether or not Clark is a mathemagician.

If Clark is a mathemagician:

   - Lois is his assistant (from (1))

   - everyone at the theater has a good time (from (2))

If Clark is NOT a mathemagican:

   (we don't know anything yet)

From (3) we also know that IF everyone in the theater has a good time, 
then Lois is not his assistant.

HOLD ON! If Clark is a mathemagician, then everyone has a good time. 
So, according to (3), if Clark is a mathemagician, then Lois is not 
his assistant. Do you see where we got this? BUT, from (1), we know 
that if Clark is a mathemagician, then Lois is his assistant.

So we have a contradiction: Lois both MUST BE (from (1)) and CAN'T BE 
(from (2)) Clark's assistant. 

But we got this contradiction because we assumed that Clark was a  
mathemagician. So he can't be one, as (4) says.

Does this make sense? And since we don't know anything if Clark is not
a mathemagician, there is no contradiction.

Thanks for your question. Please write back if you need more help.

-Doctor Jodi,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/