Alright! Let's construct the truth table step-by-step for the compound statement [tex]\(\sim( P \vee \sim Q ) \leftrightarrow (\sim P \wedge Q)\)[/tex].
First, let's break down the statement into simpler components and compute the truth values for each component, ultimately leading to the final compound statement.
Here are the steps:
1. List all possible truth values for [tex]\(P\)[/tex] and [tex]\(Q\)[/tex].
2. Compute [tex]\(\sim Q\)[/tex] because it is used within the expression.
3. Compute [tex]\(P \vee \sim Q\)[/tex].
4. Compute [tex]\(\sim(P \vee \sim Q)\)[/tex].
5. Compute [tex]\(\sim P\)[/tex].
6. Compute [tex]\(\sim P \wedge Q\)[/tex].
7. Finally, compute [tex]\(\sim( P \vee \sim Q ) \leftrightarrow (\sim P \wedge Q)\)[/tex].
Now let's create the truth table:
| [tex]\(P\)[/tex] | [tex]\(Q\)[/tex] | [tex]\(\sim Q\)[/tex] | [tex]\(P \vee \sim Q\)[/tex] | [tex]\(\sim(P \vee \sim Q)\)[/tex] | [tex]\(\sim P\)[/tex] | [tex]\(\sim P \wedge Q\)[/tex] | [tex]\(\sim( P \vee \sim Q ) \leftrightarrow (\sim P \wedge Q)\)[/tex] |
|------|------|-------------|------------------|-----------------------|------------|---------------------|-----------------------------------------------|
| T | T | F | T | F | F | F | T |
| T | F | T | T | F | F | F | T |
| F | T | F | F | T | T | T | T |
| F | F | T | T | F | T | F | F |
Let's summarize the columns:
- [tex]\(\sim Q\)[/tex]: Negation of [tex]\(Q\)[/tex].
- [tex]\(P \vee \sim Q\)[/tex]: Logical OR between [tex]\(P\)[/tex] and [tex]\(\sim Q\)[/tex].
- [tex]\(\sim(P \vee \sim Q)\)[/tex]: Negation of [tex]\(P \vee \sim Q\)[/tex].
- [tex]\(\sim P\)[/tex]: Negation of [tex]\(P\)[/tex].
- [tex]\(\sim P \wedge Q\)[/tex]: Logical AND between [tex]\(\sim P\)[/tex] and [tex]\(Q\)[/tex].
- [tex]\(\sim( P \vee \sim Q ) \leftrightarrow (\sim P \wedge Q)\)[/tex]: Biconditional between [tex]\(\sim( P \vee \sim Q )\)[/tex] and [tex]\(\sim P \wedge Q\)[/tex].
By following these steps, you can see that the truth values in the final column match the expected result of the compound statement.
