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The exercise asks to translate logical propositions into colloquial language, with:

- **p** = "ella tiene ojos azules" (she has blue eyes)
- **q** = "él tiene 45 años de edad" (he is 45 years old)

I'll now translate each logical expression:

**4.1)**  \(\neg p\)  
*No es cierto que ella tiene ojos azules.*  
(It is not true that she has blue eyes.)

**4.2)**  \(\neg q\)  
*No es cierto que él tiene 45 años de edad.*  
(It is not true that he is 45 years old.)

**4.3)**  \(p \land q\)  
*Ella tiene ojos azules y él tiene 45 años de edad.*  
(She has blue eyes, and he is 45 years old.)

**4.4)**  \(p \lor q\)  
*Ella tiene ojos azules o él tiene 45 años de edad.*  
(She has blue eyes, or he is 45 years old.)

**4.5)**  \(\neg p \lor q\)  
*Ella no tiene ojos azules o él tiene 45 años de edad.*  
(She does not have blue eyes, or he is 45 years old.)

**4.6)**  \(p \land \neg q\)  
*Ella tiene ojos azules y no es cierto que él tiene 45 años de edad.*  
(She has blue eyes, and it is not true that he is 45 years old.)

**4.7)**  \(\neg p \lor q\)  
*Ella no tiene ojos azules o él tiene 45 años de edad.*  
(She does not have blue eyes, or he is 45 years old.)  
(Note: This is the same as 4.5, due to a repeated form.)

**4.8)**  \(\neg p \land q\)  
*Ella no tiene ojos azules y él tiene 45 años de edad.*  
(She does not have blue eyes, and he is 45 years old.)

**4.9)**  \(\neg (p \Rightarrow \neg q)\)  
*No es cierto que si ella tiene ojos azules, entonces él no tiene 45 años de edad.*  
(It is not true that if she has blue eyes, then he is not 45 years old.)

**4.10)**  \(p \Leftarrow \neg q\)  
*Ella tiene ojos azules si él no tiene 45 años de edad.*  
(She has blue eyes if he is not 45 years old.)

**4.11)**  \(p \Leftrightarrow q\)  
*Ella tiene ojos azules si y solo si él tiene 45 años de edad.*  
(She has blue eyes if and only if he is 45 years old.)

**4.12)**  \(\neg (\neg p \Leftrightarrow \neg q)\)  
*No es cierto que ella no tiene ojos azules si y solo si él no tiene 45 años de edad.*  
(It is not true that she does not have blue eyes if and only if he is not 45 years old.)

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Would you like more details or have any questions about these translations?

### Related questions:
1. How do conjunctions and disjunctions differ in logic?
2. What does it mean when two propositions are biconditional?
3. How does the negation of a conditional statement affect its meaning?
4. Can logical equivalences be used to simplify complex propositions?
5. What is the difference between a tautology and a contradiction in logic?

### Tip:  
Remember that the negation of a conditional statement \(\neg(p \Rightarrow q)\) is logically equivalent to \(p \land \neg q\), meaning that the first statement is true and the second is false.

Represent logical propositions using p for 'ella tiene ojos azules' and q for 'él tiene 45 años de edad.' Translate each proposition into colloquial language.

This exercise focuses on translating logical propositions such as 'ella tiene ojos azules' and 'él tiene 45 años de edad' into colloquial language using propositional logic and logical connectives. Suitable for college-level students studying discrete mathematics or logic.

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