The dual of the statement `(p vee q) wedge (r vee s)`

Options

A. `(p wedge q) wedge (r wedge s)`
B. `(p wedge q) vee (r wedge s)`
C. `(p vee q) vee (r vee s)`
D. `(p vee q) wedge (r vee s)`

Answer

To find the dual of the statement \((p \vee q) \wedge (r \vee s)\), we follow these steps:

### Step 1: Identify the Operators
The given expression is \((p \vee q) \wedge (r \vee s)\). Here, we have:
- \(\vee\) (OR) operator
- \(\wedge\) (AND) operator

### Step 2: Apply the Duality Principle
According to the duality principle in Boolean algebra, we need to swap the AND and OR operators:
- Replace every \(\vee\) with \(\wedge\)
- Replace every \(\wedge\) with \(\vee\)

### Step 3: Apply the Changes
Applying the duality principle to our expression:
- The expression \((p \vee q) \wedge (r \vee s)\) becomes:
- Change \(\wedge\) to \(\vee\)
- Change \(\vee\) to \(\wedge\)

Thus, we get:
\[ (p \wedge q) \vee (r \wedge s) \]

### Step 4: Final Expression
The dual of the statement \((p \vee q) \wedge (r \vee s)\) is:
\[ (p \wedge q) \vee (r \wedge s) \]