If the Boolean expression `(p oplusq)wedge (~p Theta q)` is equivalent to `p wedge q`, where `oplus, Theta in {vee, wedge}`, then the ordered pair `(``oplus, Theta`)` is

Options

A. `( wedge, vee)`
B. `( vee, vee)`
C. `( wedge, wedge)`
D. `(vee, wedge)`

Answer

To solve the problem, we need to find the ordered pair \((\oplus, \Theta)\) such that the Boolean expression \((p \oplus q) \wedge (\neg p \Theta q)\) is equivalent to \(p \wedge q\). Here, \(\oplus\) and \(\Theta\) can be either conjunction (\(\wedge\)) or disjunction (\(\vee\)).

### Step 1: Understand the equivalence
We start with the expression:
\[ (p \oplus q) \wedge (\neg p \Theta q) \equiv p \wedge q \]
We need to check different combinations of \(\oplus\) and \(\Theta\) to find which one satisfies this equivalence.

### Step 2: Create a truth table for \(p \wedge q\)
The truth table for \(p \wedge q\) is as follows:

| \(p\) | \(q\) | \(p \wedge q\) |
|-------|-------|----------------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |

### Step 3: Check combinations of \(\oplus\) and \(\Theta\)

#### Case 1: \(\oplus = \wedge\), \(\Theta = \vee\)
1. Calculate \(p \wedge q\):
- This is already given in the truth table.

2. Calculate \(\neg p \vee q\):
- The truth table for \(\neg p\) is:

| \(p\) | \(\neg p\) |
|-------|------------|
| T | F |
| F | T |

- Now, calculate \(\neg p \vee q\):

| \(p\) | \(q\) | \(\neg p\) | \(\neg p \vee q\) |
|-------|-------|------------|--------------------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

3. Now calculate \((p \wedge q) \wedge (\neg p \vee q)\):

| \(p\) | \(q\) | \(p \wedge q\) | \(\neg p \vee q\) | \((p \wedge q) \wedge (\neg p \vee q)\) |
|-------|-------|----------------|--------------------|------------------------------------------|
| T | T | T | T | T |
| T | F | F | F | F |
| F | T | F | T | F |
| F | F | F | T | F |

This results in:
\[ (p \wedge q) \wedge (\neg p \vee q) \equiv p \wedge q \]

#### Case 2: \(\oplus = \vee\), \(\Theta = \wedge\)
1. Calculate \(p \vee q\):

| \(p\) | \(q\) | \(p \vee q\) |
|-------|-------|---------------|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |

2. Calculate \(\neg p \wedge q\):

| \(p\) | \(q\) | \(\neg p\) | \(\neg p \wedge q\) |
|-------|-------|------------|----------------------|
| T | T | F | F |
| T | F | F | F |
| F | T | T | T |
| F | F | T | F |

3. Now calculate \((p \vee q) \wedge (\neg p \wedge q)\):

| \(p\) | \(q\) | \(p \vee q\) | \(\neg p \wedge q\) | \((p \vee q) \wedge (\neg p \wedge q)\) |
|-------|-------|---------------|----------------------|------------------------------------------|
| T | T | T | F | F |
| T | F | T | F | F |
| F | T | T | T | T |
| F | F | F | F | F |

This does not match \(p \wedge q\).

### Conclusion
The only ordered pair that satisfies the equivalence is:
\[ (\oplus, \Theta) = (\wedge, \vee) \]