QUESTION IMAGE
Question
use a truth table to determine whether the two statements are equivalent
$p \
ightarrow \sim q$ and $\sim q \
ightarrow \sim p$
choose the correct answer below.
the statements are not equivalent.
the statements are equivalent.
Step1: List all p, q truth values
| $p$ | $q$ |
|---|---|
| T | F |
| F | T |
| F | F |
Step2: Calculate $\sim q$ and $\sim p$
| $p$ | $q$ | $\sim q$ | $\sim p$ |
|---|---|---|---|
| T | F | T | F |
| F | T | F | T |
| F | F | T | T |
Step3: Compute $p
ightarrow \sim q$
Recall $a
ightarrow b$ is F only if $a=T, b=F$.
| $p$ | $q$ | $\sim q$ | $\sim p$ | $p |
ightarrow \sim q$ |
| T | T | F | F | F |
| T | F | T | F | T |
| F | T | F | T | T |
| F | F | T | T | T |
Step4: Compute $\sim q
ightarrow \sim p$
| $p$ | $q$ | $\sim q$ | $\sim p$ | $p |
ightarrow \sim q$ | $\sim q
ightarrow \sim p$ |
| T | T | F | F | F | T |
| T | F | T | F | T | F |
| F | T | F | T | T | T |
| F | F | T | T | T | T |
Step5: Compare the two columns
The columns for $p
ightarrow \sim q$ and $\sim q
ightarrow \sim p$ do not match (rows 1 and 2 differ).
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The statements are not equivalent.