To understand and solve this logical statement problem, let's break down the information provided and analyze the choices step-by-step.
The given statement is:
- If it’s a weekend (represented as [tex]\( p \)[/tex]), I exercise (represented as [tex]\( q \)[/tex]).
- It’s not a weekend (represented as [tex]\( \neg p \)[/tex]).
- Therefore, I won’t exercise (represented as [tex]\( \neg q \)[/tex]).
### Step-by-Step Detailed Solution:
1. Identify the main logical components:
- The statement "If it’s a weekend, I exercise" can be represented as [tex]\( p \rightarrow q \)[/tex].
- "It’s not a weekend" is represented as [tex]\( \neg p \)[/tex].
- The conclusion "So, I won’t exercise" is represented as [tex]\( \neg q \)[/tex].
2. Define the logical argument:
- We start with [tex]\( (p \rightarrow q) \)[/tex] and [tex]\( \neg p \)[/tex].
- From these premises, we aim to conclude [tex]\( \neg q \)[/tex].
3. Logical Inference:
- The logical rule that applies here is Modus Tollens: If [tex]\( p \rightarrow q \)[/tex] and [tex]\( \neg p \)[/tex], then [tex]\( \neg q \)[/tex].
4. Check the provided options:
A. [tex]\((p \wedge q) \rightarrow \neg p\)[/tex]
- This statement translates to "If it’s a weekend and I exercise, then it’s not a weekend."
- This option doesn’t fit our logical progression, as it contradicts the given information.
B. [tex]\([(p \rightarrow q) \wedge p] \rightarrow \neg q\)[/tex]
- This statement translates to "If (if it’s a weekend, I exercise) and it’s a weekend, then I won’t exercise."
- This option has a flaw because it starts with the assumption that it’s a weekend [tex]\( p \)[/tex], which contradicts the given premise [tex]\( \neg p \)[/tex].
C. [tex]\([(p \rightarrow q) \wedge \neg p] \rightarrow \neg q\)[/tex]
- This statement translates to "If (if it’s a weekend, I exercise) and it’s not a weekend, then I won’t exercise."
- This is a correct representation of the given argument and follows the logic we identified.
D. [tex]\([(p \wedge q) \rightarrow \neg p] \rightarrow \neg q\)[/tex]
- This statement translates to "If (it’s a weekend and I exercise implies it’s not a weekend), then I won’t exercise."
- This statement is convoluted and does not correctly represent the logical flow we outlined.
5. Conclusion:
The correct logical statement representing the given argument is:
C. [tex]\([(p \rightarrow q) \wedge \neg p] \rightarrow \neg q\)[/tex]
Thus, the answer is:
[tex]\[ \boxed{C} \][/tex]
