Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To determine which statement is logically equivalent to the conditional statement [tex]\(\sim p \rightarrow q\)[/tex], we need to understand logical equivalences and how they can be used to transform the given statement.
### Step-by-Step Solution:
1. Original Statement Analysis:
- The given statement is [tex]\(\sim p \rightarrow q\)[/tex].
- To find an equivalent statement, we use the fact that an implication [tex]\(A \rightarrow B\)[/tex] can be rewritten using the equivalence:
[tex]\[ A \rightarrow B \equiv \sim A \vee B \][/tex]
2. Apply the Equivalence:
- For the statement [tex]\(\sim p \rightarrow q\)[/tex], treat [tex]\(\sim p\)[/tex] as [tex]\(A\)[/tex] and [tex]\(q\)[/tex] as [tex]\(B\)[/tex].
- So, [tex]\[\sim p \rightarrow q \equiv \sim (\sim p) \vee q\][/tex]
3. Simplify the Expression:
- Simplifying [tex]\(\sim (\sim p)\)[/tex] gives [tex]\(p\)[/tex].
- Thus, [tex]\[\sim p \rightarrow q \equiv p \vee q\][/tex]
4. Compare with Given Choices:
- Now we need to compare [tex]\(p \vee q\)[/tex] with the equivalent forms of the given choices:
1. [tex]\(p \rightarrow \sim q\)[/tex]
2. [tex]\(\sim p \rightarrow \sim q\)[/tex]
3. [tex]\(\sim q \rightarrow \sim p\)[/tex]
4. [tex]\(\sim q \rightarrow p\)[/tex]
5. Transform Each Choice:
- Let's find the logical equivalences for each option in terms of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
1. [tex]\(p \rightarrow \sim q\)[/tex] is equivalent to [tex]\(\sim p \vee \sim q\)[/tex]
2. [tex]\(\sim p \rightarrow \sim q\)[/tex] is equivalent to [tex]\(p \vee \sim q\)[/tex]
3. [tex]\(\sim q \rightarrow \sim p\)[/tex] is equivalent to [tex]\(q \vee p\)[/tex] (which is logically equivalent to [tex]\(p \vee q\)[/tex])
4. [tex]\(\sim q \rightarrow p\)[/tex] is equivalent to [tex]\(q \vee \sim p\)[/tex]
6. Check for Equivalence:
- We need the form [tex]\(p \vee q\)[/tex], which matches exactly with the form obtained from the given statement [tex]\(\sim p \rightarrow q\)[/tex].
From the transformed logical forms, we see that option 2:
[tex]\(\sim p \rightarrow \sim q \equiv p \vee q\)[/tex] fits our derived form.
Therefore, the statement logically equivalent to [tex]\(\sim p \rightarrow q\)[/tex] is:
[tex]\[ \boxed{\sim p \rightarrow \sim q} \][/tex]
### Step-by-Step Solution:
1. Original Statement Analysis:
- The given statement is [tex]\(\sim p \rightarrow q\)[/tex].
- To find an equivalent statement, we use the fact that an implication [tex]\(A \rightarrow B\)[/tex] can be rewritten using the equivalence:
[tex]\[ A \rightarrow B \equiv \sim A \vee B \][/tex]
2. Apply the Equivalence:
- For the statement [tex]\(\sim p \rightarrow q\)[/tex], treat [tex]\(\sim p\)[/tex] as [tex]\(A\)[/tex] and [tex]\(q\)[/tex] as [tex]\(B\)[/tex].
- So, [tex]\[\sim p \rightarrow q \equiv \sim (\sim p) \vee q\][/tex]
3. Simplify the Expression:
- Simplifying [tex]\(\sim (\sim p)\)[/tex] gives [tex]\(p\)[/tex].
- Thus, [tex]\[\sim p \rightarrow q \equiv p \vee q\][/tex]
4. Compare with Given Choices:
- Now we need to compare [tex]\(p \vee q\)[/tex] with the equivalent forms of the given choices:
1. [tex]\(p \rightarrow \sim q\)[/tex]
2. [tex]\(\sim p \rightarrow \sim q\)[/tex]
3. [tex]\(\sim q \rightarrow \sim p\)[/tex]
4. [tex]\(\sim q \rightarrow p\)[/tex]
5. Transform Each Choice:
- Let's find the logical equivalences for each option in terms of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
1. [tex]\(p \rightarrow \sim q\)[/tex] is equivalent to [tex]\(\sim p \vee \sim q\)[/tex]
2. [tex]\(\sim p \rightarrow \sim q\)[/tex] is equivalent to [tex]\(p \vee \sim q\)[/tex]
3. [tex]\(\sim q \rightarrow \sim p\)[/tex] is equivalent to [tex]\(q \vee p\)[/tex] (which is logically equivalent to [tex]\(p \vee q\)[/tex])
4. [tex]\(\sim q \rightarrow p\)[/tex] is equivalent to [tex]\(q \vee \sim p\)[/tex]
6. Check for Equivalence:
- We need the form [tex]\(p \vee q\)[/tex], which matches exactly with the form obtained from the given statement [tex]\(\sim p \rightarrow q\)[/tex].
From the transformed logical forms, we see that option 2:
[tex]\(\sim p \rightarrow \sim q \equiv p \vee q\)[/tex] fits our derived form.
Therefore, the statement logically equivalent to [tex]\(\sim p \rightarrow q\)[/tex] is:
[tex]\[ \boxed{\sim p \rightarrow \sim q} \][/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
