Question
d. Let \( p: \) I build a rocket. Let \( \mathbf{q}: \) I fly to Mars. Given the statement "If I build a rocket, then I will not fly to Mars." The logically equivalent negation of the conditional statement is \[ p \wedge \sim q \] \( O \sim p \wedge q \) \( O \sim p \wedge \sim q \) \( O p \wedge q \)
Ask by Rodriquez Colon. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The logically equivalent negation of the conditional statement "If I build a rocket, then I will not fly to Mars" is \( p \wedge q \).
Solution
We start with the conditional statement:
\[
p \to \sim q
\]
This means "If I build a rocket, then I will not fly to Mars."
The negation of the conditional is:
\[
\sim (p \to \sim q)
\]
Recall that a conditional \( p \to \sim q \) is logically equivalent to \( \sim p \lor \sim q \). Thus, we have:
\[
\sim (p \to \sim q) = \sim (\sim p \lor \sim q)
\]
Using De Morgan's Law, which states that \( \sim (A \lor B) \equiv \sim A \wedge \sim B \), we obtain:
\[
\sim (\sim p \lor \sim q) = \sim (\sim p) \wedge \sim (\sim q)
\]
Since \( \sim (\sim p) \) is equivalent to \( p \) and \( \sim (\sim q) \) is equivalent to \( q \), the expression simplifies to:
\[
p \wedge q
\]
Thus, the logically equivalent negation of the conditional statement is:
\[
p \wedge q
\]
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Simplify this solution The Deep Dive
The statement "If I build a rocket, then I will not fly to Mars" can be written logically as \( p \rightarrow \sim q \). To find the logically equivalent negation, we need to negate this entire conditional statement. The negation of \( p \rightarrow \sim q \) is equivalent to \( p \wedge q \)—meaning both that you build the rocket and that you do fly to Mars. So, from your options, the correct answer is: \( O p \wedge q \) If you want to visualize it, think of it like this: constructing a rocket is half the fun, but zooming off to Mars is the ultimate adventure we all crave!
