Elise, Fred, and Gaétan are suspected of committing a theft. We have the following information about them:

- if Gaétan is not guilty, then Fred is guilty.
- If Elise is not guilty, then Gaétan is guilty.
- If Gaétan is guilty, then Elise is also guilty.
- If Elise is guilty, then Fred is not guilty.

Question: Who is or are the guilty one(s)
The 4 propositions given above must all be true simultaneously. We will then show using propositions 2 and 3 that Elise is necessarily guilty. Let's reason by contradiction: suppose that Elise is not guilty. Then by proposition 2, Gaétan is guilty. Therefore, by proposition 3, Elise is guilty. This contradicts the initial assumption. Therefore, Elise is guilty.

Since Elise is guilty, Fred is not guilty according to proposition 4. We deduce from the contrapositive of the first implication that Gaétan is guilty.

Conclusion: the culprits of this theft are Elise and Gaétan.

## IV) Additional exercises.