Write 'T' for True and 'F' for False.
`(p vv q) ^^ q = p`.

Answer

To determine the truth value of the expression \((p \lor q) \land q = p\), we will construct a truth table and analyze the results step by step.

### Step 1: Define the Variables
Let \(p\) and \(q\) be two logical variables that can take values True (T) or False (F).

### Step 2: Construct the Truth Table
We will create a truth table with the following columns:
1. \(p\)
2. \(q\)
3. \(p \lor q\) (disjunction)
4. \((p \lor q) \land q\) (conjunction)
5. Compare \((p \lor q) \land q\) with \(p\)

The truth table will have four rows corresponding to all combinations of truth values for \(p\) and \(q\).

| \(p\) | \(q\) | \(p \lor q\) | \((p \lor q) \land q\) | \((p \lor q) \land q = p\) |
|-------|-------|---------------|--------------------------|------------------------------|
| T | T | T | T | T |
| T | F | T | F | F |
| F | T | T | T | F |
| F | F | F | F | T |

### Step 3: Fill in the Truth Values
- For \(p = T\) and \(q = T\):
- \(p \lor q = T\) (since at least one is true)
- \((p \lor q) \land q = T \land T = T\)
- \(T = T\) (True)

- For \(p = T\) and \(q = F\):
- \(p \lor q = T\)
- \((p \lor q) \land q = T \land F = F\)
- \(F = T\) (False)

- For \(p = F\) and \(q = T\):
- \(p \lor q = T\)
- \((p \lor q) \land q = T \land T = T\)
- \(T = F\) (False)

- For \(p = F\) and \(q = F\):
- \(p \lor q = F\)
- \((p \lor q) \land q = F \land F = F\)
- \(F = F\) (True)

### Step 4: Analyze the Results
Now we compare the last column with the column for \(p\):
- For \(p = T\) and \(q = T\): True
- For \(p = T\) and \(q = F\): False
- For \(p = F\) and \(q = T\): False
- For \(p = F\) and \(q = F\): True

The results show that \((p \lor q) \land q\) is not equivalent to \(p\) for all combinations of truth values. Therefore, the statement \((p \lor q) \land q = p\) is **False**.

### Final Answer
The truth value of the expression \((p \lor q) \land q = p\) is **F** (False).

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