Question

chocolate luca has a bag of miniature chocolate bars that come in two distinct types: dark and milk. luca picks a chocolate bar out of the bag. use the following statements to determine whether the statement ( sim (sim p land sim q) ) is true.
p: the chocolate bar is dark chocolate
q: the chocolate bar is milk chocolate
select choice

Explanation:

Step1: Recall Negation Rules

The negation of a statement \( p \) (denoted \( \sim p \)) is true when \( p \) is false, and vice versa. For the compound statement \( \sim (p \land \sim q) \), we first simplify using De Morgan's Law: \( \sim (p \land \sim q)=\sim p \lor q \) (since \( \sim (A \land B)=\sim A \lor \sim B \), here \( A = p \), \( B=\sim q \), so \( \sim p \lor \sim (\sim q)=\sim p \lor q \)).

Step2: Analyze \( p \) and \( q \)

• \( p \): "the chocolate bar is dark chocolate"
• \( q \): "the chocolate bar is milk chocolate"

A chocolate bar can't be both dark and milk (mutually exclusive in this context). So:

• \( \sim p \): "the chocolate bar is NOT dark chocolate" (i.e., it's milk chocolate, so \( \sim p = q \))
• \( \sim q \): "the chocolate bar is NOT milk chocolate" (i.e., it's dark chocolate, so \( \sim q = p \))

Step3: Evaluate \( \sim (p \land \sim q) \)

Substitute \( \sim q = p \) into \( p \land \sim q \): \( p \land p = p \). Then \( \sim (p \land \sim q)=\sim p \). But from \( \sim p = q \), so \( \sim (p \land \sim q)=q \). Wait, no—let's re - do with actual truth values.

Case 1: If the chocolate bar is dark ( \( p = T \), \( q = F \)):

• \( \sim q = T \)
• \( p \land \sim q = T \land T = T \)
• \( \sim (p \land \sim q)=F \)

Case 2: If the chocolate bar is milk ( \( p = F \), \( q = T \)):

• \( \sim q = F \)
• \( p \land \sim q = F \land F = F \)
• \( \sim (p \land \sim q)=T \)

So \( \sim (p \land \sim q) \) is true when the chocolate bar is milk chocolate (i.e., \( q \) is true). So the statement \( \sim (p \land \sim q) \) is true when the chocolate bar is milk chocolate, which means we need to see when \( \sim (p \land \sim q) \) holds. But the question is to determine when \( \sim (p \land \sim q) \) is true. From the above, when \( q \) is true (milk chocolate), \( \sim (p \land \sim q) \) is true.

Wait, maybe a better way: Let's use the original expression \( \sim (p \land \sim q) \). Let's compute its truth table:

\( p \)	\( q \)	\( \sim q \)	\( p \land \sim q \)	\( \sim (p \land \sim q) \)
F	T	F	F	T

So \( \sim (p \land \sim q) \) is true when \( p \) is false and \( q \) is true (i.e., the chocolate bar is milk chocolate). So the statement \( \sim (p \land \sim q) \) is true when the chocolate bar is milk chocolate ( \( q \) is true).

Answer:

The statement \( \sim (p \land \sim q) \) is true when the chocolate bar is milk chocolate (i.e., when \( q \) is true). So if we are to say when \( \sim (p \land \sim q) \) is true, the answer is when the chocolate bar is milk chocolate (corresponding to \( q \) being true).