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The question asks to symbolize the following sentence:  
*"Los soldados vencieron en el campo de batalla, pero no ganaron la guerra. Luego, si no vencieron en la batalla, no ganaron la guerra."*

Let's break this down:

1. **"Los soldados vencieron en el campo de batalla"**  
   This can be represented as proposition **p** (p = They won the battle).

2. **"No ganaron la guerra"**  
   This can be represented as **¬q** (¬q = They did not win the war).

3. **"Si no vencieron en la batalla, no ganaron la guerra"**  
   This is a conditional statement, which translates to:  
   If **not p**, then **not q** (¬p → ¬q).

The sentence combines a conjunction and a conditional:

- "They won the battle but did not win the war" means **p ∧ ¬q**.
- Then, "If they did not win the battle, they did not win the war" is **¬p → ¬q**.

### Now let's compare these elements with the options:

a) **(p → q) → (q → r)**  
   This doesn't match the structure of the sentence since it's unrelated to **p ∧ ¬q** or **¬p → ¬q**.

b) **(p → q) ∧ (q → r)**  
   Similar to option (a), but still doesn't reflect the conjunction **p ∧ ¬q** or the conditional **¬p → ¬q**.

c) **(p ∧ ¬q) → (¬p → ¬q)**  
   This matches the structure:  
   - **p ∧ ¬q**: They won the battle but did not win the war.  
   - **(¬p → ¬q)**: If they did not win the battle, they did not win the war.

d) **(p ∧ q) → (q → ¬r)**  
   This doesn't match the sentence.

e) **(p ∧ ¬q) → (¬p ∧ q)**  
   This is incorrect because it suggests that if they won the battle but not the war, they didn’t win the battle but won the war, which contradicts the statement.

### Correct Answer:
The correct option is **c)**.

Would you like more details, or have any questions?  
Here are 5 related questions for further exploration:

1. How are logical connectors used in symbolic logic?
2. What is the difference between conjunction and implication in logic?
3. Can you explain the concept of contraposition in logical statements?
4. How do negations affect the truth value of logical propositions?
5. How can truth tables be used to verify logical equivalences?

**Tip**: When dealing with logical translations, carefully identify each part of the sentence and its logical connector before symbolizing.

Simbolizar: 'Los soldados vencieron en el campo de batalla, pero no ganaron la guerra. Luego si no vencieron en la batalla, no ganaron la guerra.'

Explore symbolic logic with a sentence about soldiers winning the battle but not the war. Learn how to symbolize statements using conjunctions, negations, and implications, with solutions comparing different logical structures. Suitable for college-level logic students.

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