The dual of `(p vv q) ^^ r` is :

Options

A. `(p ^^ q) ^^ r`
B. `(p ^^ q) vv r`
C. `(p vv q) vv r`
D. `p ^^ r`

Answer

To find the dual of the expression `(p ∨ q) ∧ r`, we will follow the rules of duality in propositional logic. The duality principle states that we can replace conjunctions (∧) with disjunctions (∨) and disjunctions (∨) with conjunctions (∧).

### Step-by-Step Solution:

1. **Identify the Expression**: The given expression is `(p ∨ q) ∧ r`.

2. **Apply the Duality Principle**:
- Replace every conjunction (∧) with disjunction (∨).
- Replace every disjunction (∨) with conjunction (∧).

3. **Transform the Expression**:
- The expression `(p ∨ q)` contains a disjunction, which will be replaced by a conjunction. Thus, it becomes `(p ∧ q)`.
- The conjunction `∧ r` will be replaced by a disjunction. Thus, it becomes `∨ r`.

4. **Combine the Transformed Parts**:
- After applying the transformations, we get the dual expression: `(p ∧ q) ∨ r`.

5. **Final Answer**: The dual of `(p ∨ q) ∧ r` is `(p ∧ q) ∨ r`.