Questions: Construct a truth table for the statement. [ sim(q rightarrow sim p) ] Complete the truth table below. p q q rightarrow sim p sim(q rightarrow sim p) ------------ T T mathbfv barnabla T F nabla nabla F T nabla nabla F F nabla nabla

Construct a truth table for the statement.
[
sim(q rightarrow sim p)
]

Complete the truth table below.

p q q rightarrow sim p sim(q rightarrow sim p)
------------
T T mathbfv barnabla
T F nabla nabla
F T nabla nabla
F F nabla nabla

Transcript text: Construct a truth table for the statement. \[ \sim(q \rightarrow \sim p) \] Complete the truth table below. \begin{tabular}{|c|c|c|c|} \hline$p$ & $q$ & $q \rightarrow \sim p$ & $\sim(q \rightarrow \sim p)$ \\ \hline$T$ & $T$ & $\mathbf{v}$ & $\bar{\nabla}$ \\ \hline$T$ & $F$ & $\nabla$ & $\nabla$ \\ \hline$F$ & $T$ & $\nabla$ & $\nabla$ \\ \hline$F$ & $F$ & $\nabla$ & $\nabla$ \\ \hline \end{tabular}

Solution

Solution Steps

To construct the truth table for the statement $\sim(q \rightarrow \sim p)$, we need to evaluate the logical expression step-by-step for all possible truth values of $p$ and $q$. The steps are as follows:

Determine the truth value of $\sim p$ for each combination of $p$ and $q$.
Determine the truth value of $q \rightarrow \sim p$ using the implication truth table.
Determine the truth value of $\sim(q \rightarrow \sim p)$ by negating the result from step 2.

Step 1: Determine the truth values of $\sim p$

For each combination of $p$ and $q$, we first determine the truth value of $\sim p$:

When $p = \text{True}$, $\sim p = \text{False}$
When $p = \text{False}$, $\sim p = \text{True}$

Step 2: Determine the truth values of $q \rightarrow \sim p$

Next, we use the implication truth table to determine $q \rightarrow \sim p$:

$q \rightarrow \sim p$ is $\text{False}$ only when $q = \text{True}$ and $\sim p = \text{False}$
Otherwise, $q \rightarrow \sim p$ is $\text{True}$

Step 3: Determine the truth values of $\sim(q \rightarrow \sim p)$

Finally, we negate the result from Step 2 to get $\sim(q \rightarrow \sim p)$:

$\sim(q \rightarrow \sim p)$ is $\text{True}$ when $q \rightarrow \sim p$ is $\text{False}$
$\sim(q \rightarrow \sim p)$ is $\text{False}$ when $q \rightarrow \sim p$ is $\text{True}$

Final Answer

\[ \boxed{ \begin{array}{|c|c|c|c|} \hline p & q & q \rightarrow \sim p & \sim(q \rightarrow \sim p) \\ \hline T & T & F & T \\ \hline T & F & T & F \\ \hline F & T & T & F \\ \hline F & F & T & F \\ \hline \end{array} } \]

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