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Indirect Proofs

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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!
```

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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
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/
```
Associated Topics:
High School Logic

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