1) For each of the following decide whether it is a proposition or not, and if
it is, indicate whether it is true or not.
(i) 15 is a positive number,
(ii) The Earth is flat,
(iii) x
2 ≥ 0,
(iv) x
2 ≥ 0 for every real number x,
(v) Shakespeare wrote the play Hamlet,
(vi) Hamlet is the best play ever written,
(vii) The next sentence is true,
(viii) The previous sentence is false
2) Let p, q and r denote the propositions
p: The bats are flying around,
q: The vampires are out of their coffins,
r: It is daytime.
Write the following in symbolic form
(i) It is nighttime and the vampires are in their coffins,
(ii) The bats are flying around and either the vampires are out of their
coffins or it is daytime,
(iii) It is not the case that either it is daytime or the vampires are in their
coffins,
(iv) Either it is daytime and the bats are flying around or the bats are
not flying around and the vampires are out of their coffins.


### 1) **Decide whether the statement is a proposition and, if so, whether it is true or false:**

(i) **15 is a positive number:**  
- **Proposition?** Yes  
- **True or False?** True

(ii) **The Earth is flat:**  
- **Proposition?** Yes  
- **True or False?** False

(iii) **\(x^2 \geq 0\):**  
- **Proposition?** No, because it is an open sentence. It depends on the value of \(x\). 

(iv) **\(x^2 \geq 0\) for every real number \(x\):**  
- **Proposition?** Yes  
- **True or False?** True, since the square of any real number is non-negative.

(v) **Shakespeare wrote the play *Hamlet*:**  
- **Proposition?** Yes  
- **True or False?** True

(vi) **Hamlet is the best play ever written:**  
- **Proposition?** No, this is subjective and based on opinion.

(vii) **The next sentence is true:**  
- **Proposition?** No, because it's part of a paradox when combined with the next sentence.

(viii) **The previous sentence is false:**  
- **Proposition?** No, together with (vii), this forms a paradox and is not a valid proposition.

---

### 2) **Translate the statements into symbolic form using the given propositions:**

Let:
- \(p\): The bats are flying around,
- \(q\): The vampires are out of their coffins,
- \(r\): It is daytime.

(i) **It is nighttime and the vampires are in their coffins:**  
- Symbolic form: \(\neg r \land \neg q\)  
  (If it is nighttime, then it is not daytime, and the vampires are in their coffins, so \(q\) is false.)

(ii) **The bats are flying around and either the vampires are out of their coffins or it is daytime:**  
- Symbolic form: \(p \land (q \lor r)\)

(iii) **It is not the case that either it is daytime or the vampires are in their coffins:**  
- Symbolic form: \(\neg (r \lor \neg q)\)

(iv) **Either it is daytime and the bats are flying around or the bats are not flying around and the vampires are out of their coffins:**  
- Symbolic form: \((r \land p) \lor (\neg p \land q)\)

---

### Would you like more details or have any questions?

### Five related questions to expand your understanding:
1. What distinguishes a proposition from an open sentence in logic?
2. Can a proposition be neither true nor false, and why?
3. How would you symbolize the statement "If it is nighttime, then the bats are not flying around"?
4. What is a paradox in logic, and why are statements (vii) and (viii) considered paradoxical?
5. How would the symbolic expressions change if we assume vampires are out only at night?

### **Tip:**  
When writing symbolic logic, remember that "and" corresponds to \(\land\), "or" to \(\lor\), and "not" to \(\neg\). These operators are fundamental in forming clear logical statements!

1) For each of the following decide whether it is a proposition or not, and if it is, indicate whether it is true or not.
(i) 15 is a positive number,
(ii) The Earth is flat,
(iii) x^2 ≥ 0,
(iv) x^2 ≥ 0 for every real number x,
(v) Shakespeare wrote the play Hamlet,
(vi) Hamlet is the best play ever written,
(vii) The next sentence is true,
(viii) The previous sentence is false
2) Let p, q and r denote the propositions p: The bats are flying around, q: The vampires are out of their coffins, r: It is daytime. Write the following in symbolic form (i) It is nighttime and the vampires are in their coffins, (ii) The bats are flying around and either the vampires are out of their coffins or it is daytime, (iii) It is not the case that either it is daytime or the vampires are in their coffins, (iv) Either it is daytime and the bats are flying around or the bats are not flying around and the vampires are out of their coffins.

Explore the fundamentals of propositional logic by determining whether statements are propositions and translating them into symbolic form. This problem covers logical operators such as AND (∧), OR (∨), and NOT (¬), as well as how to analyze statements and propositions. Suitable for students in Grades 9-12.

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