Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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]
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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
