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Let's solve the problems in the image you've provided. The text involves propositional logic, formalization of sentences, truth tables, and equivalences.

### Exercise 1: **Formalize the propositions**

a) **Federico se irá a las Fiji o a las Seychelles y sólo si le toca la lotería y no se arruina en la ruleta.**
   - Let: 
     - \( F \): Federico goes to Fiji
     - \( S \): Federico goes to Seychelles
     - \( L \): Federico wins the lottery
     - \( R \): Federico ruins himself in roulette
   - Formalization: \[(F \lor S) \land (L \land \neg R)\]

b) **El Hombre Lobo es un invento, y si lo mismo ocurre con Papá Noel, entonces los niños son engañados.**
   - Let:
     - \( W \): The werewolf is an invention
     - \( P \): Santa Claus is an invention
     - \( N \): Children are deceived
   - Formalization: \[W \land (P \rightarrow N)\]

c) **Cuando la bolsa baja mucho, entonces es conveniente comprar; y cuando la bolsa sube mucho, también es conveniente comprar.**
   - Let:
     - \( B_b \): The stock market drops significantly
     - \( B_s \): The stock market rises significantly
     - \( C \): It is advisable to buy
   - Formalization: \[(B_b \rightarrow C) \land (B_s \rightarrow C)\]

d) **Si viene en tren, llegará antes de las seis. Si viene en coche, llegará antes de las seis. Luego, tanto si viene en tren como si viene en coche, llegará antes de las seis.**
   - Let:
     - \( T \): Comes by train
     - \( C \): Comes by car
     - \( A \): Arrives before six
   - Formalization: \[(T \rightarrow A) \land (C \rightarrow A) \land ((T \lor C) \rightarrow A)\]

e) **María y Juana son hermanas y trabajan pero no ahorran.**
   - Let:
     - \( M \): María and Juana are sisters
     - \( T \): They work
     - \( A \): They save
   - Formalization: \[M \land T \land \neg A\]

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### Exercise 2: **How many variables are in the table? Is it a tautology?**
   - **Statement:** 
   "Si un animal fabuloso se enfada, te quedas paralizado del susto; y si te quedas paralizado del susto, entonces no puedes sino apelar a su bondad y así no ser engullido."
   - This statement can be formalized using propositional variables:
     - Let \( E \): The animal gets angry
     - Let \( P \): You are paralyzed by fear
     - Let \( A \): You appeal to its kindness
     - Let \( G \): You are not swallowed
     - Formalization: \[(E \rightarrow P) \land (P \rightarrow (A \land G))\]
   - We have four variables: \( E \), \( P \), \( A \), \( G \).
   - To determine if it’s a **tautology**, we would need to check the truth table. However, from the structure, this is **not necessarily a tautology**, as it only holds under specific conditions (e.g., if the animal does get angry).

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### Exercise 3: **Determine whether the systems are equivalent (truth tables or demonstrations).**

The given systems are:

a) \((q \rightarrow p) \land (q \land \neg p)\)
b) \(\neg (q \land \neg p) \equiv p \rightarrow \neg q\)
c) \((p \lor q) \land (\neg p \lor \neg q) \equiv (p \lor q) \rightarrow (\neg p \lor q)\)

We can use **truth tables** or **logical equivalences** to determine whether these statements are equivalent. Do you want me to construct truth tables for each system to check their equivalence?

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Let me know if you'd like further clarifications or details on any of these exercises!

Resuelva usted los siguientes ejercicios de formalización de proposiciones, tabla de verdad y equivalencias lógicas.

This problem set covers formalization of logical propositions, constructing truth tables, and determining logical equivalences. Key concepts include implications, conjunctions, and disjunctions, utilizing propositional logic for problem-solving.

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