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Answer

To solve the expression `(p ∨ q) ∧ q`, we will create a truth table to evaluate the expression step by step.

### Step 1: Identify the Variables
We have two variables:
- \( P \)
- \( Q \)

### Step 2: Create the Truth Table Structure
We will create a truth table with the following columns:
1. \( P \)
2. \( Q \)
3. \( P ∨ Q \) (Disjunction)
4. \( (P ∨ Q) ∧ Q \) (Conjunction)

### Step 3: Fill in the Truth Values
The possible truth values for \( P \) and \( Q \) are:
- True (T)
- False (F)

Thus, we will have the following combinations for \( P \) and \( Q \):
1. \( P = T, Q = T \)
2. \( P = T, Q = F \)
3. \( P = F, Q = T \)
4. \( P = F, Q = F \)

### Step 4: Calculate \( P ∨ Q \)
Now we will calculate the values for \( P ∨ Q \):
- When \( P = T \) and \( Q = T \), \( P ∨ Q = T \)
- When \( P = T \) and \( Q = F \), \( P ∨ Q = T \)
- When \( P = F \) and \( Q = T \), \( P ∨ Q = T \)
- When \( P = F \) and \( Q = F \), \( P ∨ Q = F \)

### Step 5: Calculate \( (P ∨ Q) ∧ Q \)
Now we will calculate the values for \( (P ∨ Q) ∧ Q \):
- When \( P = T \) and \( Q = T \), \( (P ∨ Q) ∧ Q = T ∧ T = T \)
- When \( P = T \) and \( Q = F \), \( (P ∨ Q) ∧ Q = T ∧ F = F \)
- When \( P = F \) and \( Q = T \), \( (P ∨ Q) ∧ Q = T ∧ T = T \)
- When \( P = F \) and \( Q = F \), \( (P ∨ Q) ∧ Q = F ∧ F = F \)

### Step 6: Summarize the Truth Table
Now we can summarize our truth table:

| P | Q | P ∨ Q | (P ∨ Q) ∧ Q |
|-------|-------|-------|-------------|
| T | T | T | T |
| T | F | T | F |
| F | T | T | T |
| F | F | F | F |

### Step 7: Analyze the Results
Now we can compare the column for \( (P ∨ Q) ∧ Q \) with the column for \( Q \):
- The values of \( (P ∨ Q) ∧ Q \) are: T, F, T, F
- The values of \( Q \) are: T, F, T, F

Since the columns for \( (P ∨ Q) ∧ Q \) and \( Q \) are identical, we conclude that:
\[ (P ∨ Q) ∧ Q \equiv Q \]

### Final Answer
Thus, the expression `(p ∨ q) ∧ q` is equivalent to \( Q \).

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