Questions: Construct a truth table for the statement. [ -p rightarrow(q wedge r) ] Fill in the blanks for the missing values in the table. p q r -p q^ r -p rightarrow(q wedge r) ------------------ T T T V v v T T F v v v T F T v checkmark v T F F 7 v v F T T v v 7 F T F v checkmark nabla F F T v nabla v F F F checkmark v 7

Construct a truth table for the statement.
[
-p rightarrow(q wedge r)
]

Fill in the blanks for the missing values in the table.

p q r -p q^ r -p rightarrow(q wedge r)
------------------
T T T V v v
T T F v v v
T F T v checkmark v
T F F 7 v v
F T T v v 7
F T F v checkmark nabla
F F T v nabla v
F F F checkmark v 7

Transcript text: Construct a truth table for the statement. \[ -p \rightarrow(q \wedge r) \] Fill in the blanks for the missing values in the table. \begin{tabular}{|c|c|c|c|c|c|c|} \hline p & q & r & -p & q^ r & \multicolumn{2}{|l|}{$-p \rightarrow(q \wedge r)$} \\ \hline T & T & T & V & v & v & \ \\ \hline T & T & F & $v$ & v & v & \ \\ \hline T & F & T & $v$ & $\checkmark$ & v & , \\ \hline T & F & F & 7 & v & v & \\ \hline F & T & T & $v$ & v & 7 & \\ \hline F & T & F & $v$ & $\checkmark$ & $\nabla$ & \\ \hline F & F & T & v & $\nabla$ & v & \\ \hline F & F & F & $\checkmark$ & v & 7 & \\ \hline \end{tabular}

Solution

Solution Steps

To construct a truth table for the statement $-p \rightarrow (q \wedge r)$, we need to evaluate each component of the statement for all possible truth values of $p$, $q$, and $r$. Specifically, we need to:

Determine the truth value of $-p$ (the negation of $p$).
Determine the truth value of $q \wedge r$ (the conjunction of $q$ and $r$).
Determine the truth value of $-p \rightarrow (q \wedge r)$ (the implication from $-p$ to $q \wedge r$).

Step 1: Construct the Truth Table

We evaluate the logical statement $-p \rightarrow (q \wedge r)$ for all combinations of truth values for $p$, $q$, and $r$. The possible combinations are:

$ (T, T, T) $
$ (T, T, F) $
$ (T, F, T) $
$ (T, F, F) $
$ (F, T, T) $
$ (F, T, F) $
$ (F, F, T) $
$ (F, F, F) $

Step 2: Calculate Intermediate Values

For each combination, we calculate:

$ -p $
$ q \wedge r $
$ -p \rightarrow (q \wedge r) $

The results for each combination are as follows:

For $ (T, T, T) $: $ -p = F $, $ q \wedge r = T $, $ -p \rightarrow (q \wedge r) = T $
For $ (T, T, F) $: $ -p = F $, $ q \wedge r = F $, $ -p \rightarrow (q \wedge r) = T $
For $ (T, F, T) $: $ -p = F $, $ q \wedge r = F $, $ -p \rightarrow (q \wedge r) = T $
For $ (T, F, F) $: $ -p = F $, $ q \wedge r = F $, $ -p \rightarrow (q \wedge r) = T $
For $ (F, T, T) $: $ -p = T $, $ q \wedge r = T $, $ -p \rightarrow (q \wedge r) = T $
For $ (F, T, F) $: $ -p = T $, $ q \wedge r = F $, $ -p \rightarrow (q \wedge r) = F $
For $ (F, F, T) $: $ -p = T $, $ q \wedge r = F $, $ -p \rightarrow (q \wedge r) = F $
For $ (F, F, F) $: $ -p = T $, $ q \wedge r = F $, $ -p \rightarrow (q \wedge r) = F $

Step 3: Summarize the Truth Table

The complete truth table is as follows:

\[ \begin{array}{|c|c|c|c|c|c|} \hline p & q & r & -p & q \wedge r & -p \rightarrow (q \wedge r) \\ \hline T & T & T & F & T & T \\ T & T & F & F & F & T \\ T & F & T & F & F & T \\ T & F & F & F & F & T \\ F & T & T & T & T & T \\ F & T & F & T & F & F \\ F & F & T & T & F & F \\ F & F & F & T & F & F \\ \hline \end{array} \]

Final Answer

The truth table for the statement $-p \rightarrow (q \wedge r)$ is complete, with the final column indicating the truth values of the implication for each combination of $p$, $q$, and $r$.

$\boxed{\text{Truth table completed successfully.}}$

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