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].
### 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].
