Certainly! Let's construct the truth table for the compound statement [tex]\((p \wedge \neg q) \wedge r\)[/tex].
We'll start with the various possible truth values for [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex]:
1. When [tex]\(p = \text{True}, q = \text{True}, r = \text{True}\)[/tex]
2. When [tex]\(p = \text{True}, q = \text{False}, r = \text{False}\)[/tex]
3. When [tex]\(p = \text{False}, q = \text{True}, r = \text{True}\)[/tex]
4. When [tex]\(p = \text{False}, q = \text{False}, r = \text{False}\)[/tex]
Let's break down the steps for each row:
### Step 1: Compute [tex]\(\neg q\)[/tex]
- [tex]\(\neg \text{True} = \text{False}\)[/tex]
- [tex]\(\neg \text{False} = \text{True}\)[/tex]
### Step 2: Compute [tex]\(p \wedge \neg q\)[/tex]
The logical AND ([tex]\(\wedge\)[/tex]) operation returns True only if both operands are True.
### Step 3: Compute [tex]\((p \wedge \neg q) \wedge r\)[/tex]
Again, the logical AND ([tex]\(\wedge\)[/tex]) operation returns True only if both operands are True.
We'll follow these steps to fill in our truth table:
| [tex]\(p\)[/tex] | [tex]\(q\)[/tex] | [tex]\(r\)[/tex] | [tex]\(\neg q\)[/tex] | [tex]\(p \wedge \neg q\)[/tex] | [tex]\((p \wedge \neg q) \wedge r\)[/tex] |
|-------|-------|-------|-----------|----------------------|------------------------------|
| True | True | True | False | False | False |
| True | False | False | True | True | False |
| False | True | True | False | False | False |
| False | False | False | True | False | False |
Thus, the completed truth table for the compound statement [tex]\((p \wedge \neg q) \wedge r\)[/tex] is:
| [tex]\(p\)[/tex] | [tex]\(q\)[/tex] | [tex]\(r\)[/tex] | [tex]\(\neg q\)[/tex] | [tex]\(p \wedge \neg q\)[/tex] | [tex]\((p \wedge \neg q) \wedge r\)[/tex] |
|-------|-------|-------|-----------|----------------------|------------------------------|
| True | True | True | False | False | False |
| True | False | False | True | True | False |
| False | True | True | False | False | False |
| False | False | False | True | False | False |
This truth table reflects all the possible values of [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex] along with their intermediate and final results.
