Write the statement in symbolic form and construct a truth table.

"It is false that the vehicle is green, or the vehicle is not an SUV."

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
$P$ & $Q$ & $\neg P \lor \neg Q$ \\
\hline
T & T & F \\
\hline
T & F & T \\
\hline
F & T & T \\
\hline
F & F & T \\
\hline
\end{tabular}
\][/tex]



Answer :

Let's break down the problem into smaller steps and construct the truth table as requested.

### Statement
The statement given is: "It is false that the vehicle is green, or the vehicle is not an SUV."

### Symbolic Form
Let:
- [tex]\( P \)[/tex] represent "The vehicle is green"
- [tex]\( Q \)[/tex] represent "The vehicle is an SUV"

The symbolic form of the given statement is:
[tex]\[ \neg P \lor \neg Q \][/tex]

### Truth Table Construction
We will now create the truth table for [tex]\( \neg P \lor \neg Q \)[/tex]:

1. List all possible combinations of [tex]\( P \)[/tex] (True or False) and [tex]\( Q \)[/tex] (True or False).
2. Calculate the negation of [tex]\( P \)[/tex] (i.e., [tex]\( \neg P \)[/tex]).
3. Calculate the negation of [tex]\( Q \)[/tex] (i.e., [tex]\( \neg Q \)[/tex]).
4. Calculate [tex]\( \neg P \lor \neg Q \)[/tex] for each combination.

Here is the step-by-step construction:

[tex]\[ \begin{array}{|c|c|c|c|c|} \hline P & Q & \neg P & \neg Q & \neg P \lor \neg Q \\ \hline \text{True} & \text{True} & \text{False} & \text{False} & \text{False} \\ \hline \text{True} & \text{False} & \text{False} & \text{True} & \text{True} \\ \hline \text{False} & \text{True} & \text{True} & \text{False} & \text{True} \\ \hline \text{False} & \text{False} & \text{True} & \text{True} & \text{True} \\ \hline \end{array} \][/tex]

### Explanation
1. First row:
- [tex]\( P \)[/tex] is True and [tex]\( Q \)[/tex] is True
- [tex]\( \neg P \)[/tex] is False and [tex]\( \neg Q \)[/tex] is False
- [tex]\( \neg P \lor \neg Q \)[/tex] is False

2. Second row:
- [tex]\( P \)[/tex] is True and [tex]\( Q \)[/tex] is False
- [tex]\( \neg P \)[/tex] is False and [tex]\( \neg Q \)[/tex] is True
- [tex]\( \neg P \lor \neg Q \)[/tex] is True

3. Third row:
- [tex]\( P \)[/tex] is False and [tex]\( Q \)[/tex] is True
- [tex]\( \neg P \)[/tex] is True and [tex]\( \neg Q \)[/tex] is False
- [tex]\( \neg P \lor \neg Q \)[/tex] is True

4. Fourth row:
- [tex]\( P \)[/tex] is False and [tex]\( Q \)[/tex] is False
- [tex]\( \neg P \)[/tex] is True and [tex]\( \neg Q \)[/tex] is True
- [tex]\( \neg P \lor \neg Q \)[/tex] is True

So, here is your detailed truth table with the final column calculated:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline P & Q & \neg P & \neg Q & \neg P \lor \neg Q \\ \hline \text{True} & \text{True} & \text{False} & \text{False} & \text{False} \\ \hline \text{True} & \text{False} & \text{False} & \text{True} & \text{True} \\ \hline \text{False} & \text{True} & \text{True} & \text{False} & \text{True} \\ \hline \text{False} & \text{False} & \text{True} & \text{True} & \text{True} \\ \hline \end{array} \][/tex]

This completes the truth table for the statement [tex]\(\neg P \lor \neg Q\)[/tex].