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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]
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