Answer :
To determine which statement is logically equivalent to the conditional statement [tex]\(\sim p \rightarrow q\)[/tex], let's analyze the statements given.
### Step 1: Rewrite the Given Statement
The conditional statement [tex]\(\sim p \rightarrow q\)[/tex] can be written using its equivalence in logical terms.
- [tex]\(\sim p \rightarrow q\)[/tex] is logically equivalent to [tex]\(p \lor q\)[/tex]. This follows from the fact that the implication [tex]\(\sim p \rightarrow q\)[/tex] is false only when [tex]\(\sim p\)[/tex] is true and [tex]\(q\)[/tex] is false, making the logical disjunction [tex]\(p \lor q\)[/tex] equivalent.
### Step 2: Analyze the Given Statements
Now we will analyze each of the given answer choices to see which one is logically equivalent to [tex]\(p \lor q\)[/tex]:
1. [tex]\(p \rightarrow \sim q\)[/tex]
- The statement [tex]\(p \rightarrow \sim q\)[/tex] is equivalent to [tex]\(\sim p \lor \sim q\)[/tex].
- This is not equivalent to [tex]\(p \lor q\)[/tex].
2. [tex]\(\sim p \rightarrow \sim q\)[/tex]
- The statement [tex]\(\sim p \rightarrow \sim q\)[/tex] is equivalent to [tex]\(p \lor \sim q\)[/tex].
- This is not equivalent to [tex]\(p \lor q\)[/tex].
3. [tex]\(\sim q \rightarrow \sim p\)[/tex]
- The statement [tex]\(\sim q \rightarrow \sim p\)[/tex] is equivalent to [tex]\(q \lor \sim p\)[/tex].
- This is not equivalent to [tex]\(p \lor q\)[/tex].
4. [tex]\(\sim q \rightarrow p\)[/tex]
- The statement [tex]\(\sim q \rightarrow p\)[/tex] is equivalent to [tex]\(q \lor p\)[/tex].
- Notice that [tex]\(q \lor p\)[/tex] is the same as [tex]\(p \lor q\)[/tex], as the disjunction operator is commutative.
### Step 3: Conclusion
Given this analysis, the statement that is logically equivalent to [tex]\(\sim p \rightarrow q\)[/tex] is [tex]\(\sim q \rightarrow p\)[/tex].
So, the correct choice is:
[tex]\[ \boxed{4} \][/tex]
### Step 1: Rewrite the Given Statement
The conditional statement [tex]\(\sim p \rightarrow q\)[/tex] can be written using its equivalence in logical terms.
- [tex]\(\sim p \rightarrow q\)[/tex] is logically equivalent to [tex]\(p \lor q\)[/tex]. This follows from the fact that the implication [tex]\(\sim p \rightarrow q\)[/tex] is false only when [tex]\(\sim p\)[/tex] is true and [tex]\(q\)[/tex] is false, making the logical disjunction [tex]\(p \lor q\)[/tex] equivalent.
### Step 2: Analyze the Given Statements
Now we will analyze each of the given answer choices to see which one is logically equivalent to [tex]\(p \lor q\)[/tex]:
1. [tex]\(p \rightarrow \sim q\)[/tex]
- The statement [tex]\(p \rightarrow \sim q\)[/tex] is equivalent to [tex]\(\sim p \lor \sim q\)[/tex].
- This is not equivalent to [tex]\(p \lor q\)[/tex].
2. [tex]\(\sim p \rightarrow \sim q\)[/tex]
- The statement [tex]\(\sim p \rightarrow \sim q\)[/tex] is equivalent to [tex]\(p \lor \sim q\)[/tex].
- This is not equivalent to [tex]\(p \lor q\)[/tex].
3. [tex]\(\sim q \rightarrow \sim p\)[/tex]
- The statement [tex]\(\sim q \rightarrow \sim p\)[/tex] is equivalent to [tex]\(q \lor \sim p\)[/tex].
- This is not equivalent to [tex]\(p \lor q\)[/tex].
4. [tex]\(\sim q \rightarrow p\)[/tex]
- The statement [tex]\(\sim q \rightarrow p\)[/tex] is equivalent to [tex]\(q \lor p\)[/tex].
- Notice that [tex]\(q \lor p\)[/tex] is the same as [tex]\(p \lor q\)[/tex], as the disjunction operator is commutative.
### Step 3: Conclusion
Given this analysis, the statement that is logically equivalent to [tex]\(\sim p \rightarrow q\)[/tex] is [tex]\(\sim q \rightarrow p\)[/tex].
So, the correct choice is:
[tex]\[ \boxed{4} \][/tex]
