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Construct the truth table for the compound statement [tex]\((p \wedge \sim q) \wedge r\)[/tex].

Fill in the empty values of the truth table. Start with the first four rows.

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

Sagot :

GinnyAnsweridn
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.
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