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.

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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: $(r \land p) \lor (\neg p \land q)$

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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!