Answer :
To determine which statement is logically equivalent to the given conditional statement [tex]\(\sim p \rightarrow q\)[/tex], we'll use logical equivalences, specifically focusing on the concept of contrapositives.
A conditional statement of the form [tex]\(A \rightarrow B\)[/tex] is logically equivalent to its contrapositive, [tex]\(\sim B \rightarrow \sim A\)[/tex]. This means that the truth value of the conditional statement remains the same when we switch the implication around and negate both parts.
Given the statement [tex]\(\sim p \rightarrow q\)[/tex]:
1. Identify the components of the statement:
- [tex]\(A\)[/tex] in this case is [tex]\(\sim p\)[/tex].
- [tex]\(B\)[/tex] in this case is [tex]\(q\)[/tex].
2. Form the contrapositive of [tex]\(\sim p \rightarrow q\)[/tex]:
- The contrapositive is obtained by negating both parts and reversing the implication.
- So, we negate [tex]\(q\)[/tex], which gives [tex]\(\sim q\)[/tex].
- We negate [tex]\(\sim p\)[/tex], which simplifies to [tex]\(p\)[/tex].
- The contrapositive of [tex]\(\sim p \rightarrow q\)[/tex] is then [tex]\(\sim q \rightarrow p\)[/tex].
3. Verify the options provided to identify the equivalent statement:
- [tex]\(p \rightarrow \sim q\)[/tex]
- [tex]\(\sim p \rightarrow \sim q\)[/tex]
- [tex]\(\sim q \rightarrow \sim p\)[/tex]
- [tex]\(\sim q \rightarrow p\)[/tex]
Among these, the statement that matches the contrapositive we derived, [tex]\(\sim q \rightarrow p\)[/tex], is:
[tex]\(\sim q \rightarrow \sim p\)[/tex]
Therefore, the statement that is logically equivalent to [tex]\(\sim p \rightarrow q\)[/tex] is:
[tex]\(\sim q \rightarrow \sim p\)[/tex]
A conditional statement of the form [tex]\(A \rightarrow B\)[/tex] is logically equivalent to its contrapositive, [tex]\(\sim B \rightarrow \sim A\)[/tex]. This means that the truth value of the conditional statement remains the same when we switch the implication around and negate both parts.
Given the statement [tex]\(\sim p \rightarrow q\)[/tex]:
1. Identify the components of the statement:
- [tex]\(A\)[/tex] in this case is [tex]\(\sim p\)[/tex].
- [tex]\(B\)[/tex] in this case is [tex]\(q\)[/tex].
2. Form the contrapositive of [tex]\(\sim p \rightarrow q\)[/tex]:
- The contrapositive is obtained by negating both parts and reversing the implication.
- So, we negate [tex]\(q\)[/tex], which gives [tex]\(\sim q\)[/tex].
- We negate [tex]\(\sim p\)[/tex], which simplifies to [tex]\(p\)[/tex].
- The contrapositive of [tex]\(\sim p \rightarrow q\)[/tex] is then [tex]\(\sim q \rightarrow p\)[/tex].
3. Verify the options provided to identify the equivalent statement:
- [tex]\(p \rightarrow \sim q\)[/tex]
- [tex]\(\sim p \rightarrow \sim q\)[/tex]
- [tex]\(\sim q \rightarrow \sim p\)[/tex]
- [tex]\(\sim q \rightarrow p\)[/tex]
Among these, the statement that matches the contrapositive we derived, [tex]\(\sim q \rightarrow p\)[/tex], is:
[tex]\(\sim q \rightarrow \sim p\)[/tex]
Therefore, the statement that is logically equivalent to [tex]\(\sim p \rightarrow q\)[/tex] is:
[tex]\(\sim q \rightarrow \sim p\)[/tex]
