The dual of `p implies q` is:

Options

A. `~| p vv q`
B. `p vv q`
C. `p ^^ q`
D. `~| p ^^ q`

Answer

To find the dual of the statement "P implies Q," we can follow these steps:

### Step-by-Step Solution:

1. **Understand the implication**: The statement "P implies Q" can be rewritten in terms of logical operations. The implication can be expressed as:
\[ P \implies Q \equiv \neg P \lor Q \]
This means "P implies Q" is equivalent to "not P or Q."

2. **Identify the dual**: The dual of a logical expression is obtained by swapping conjunctions (AND, ∧) with disjunctions (OR, ∨) and vice versa, while keeping negations (¬) unchanged.

3. **Apply duality**: Now, we apply the duality transformation to the expression \(\neg P \lor Q\):
- The disjunction (OR, ∨) becomes a conjunction (AND, ∧).
- Therefore, the dual of \(\neg P \lor Q\) is:
\[ \neg P \land Q \]

4. **Conclusion**: Thus, the dual of "P implies Q" is:
\[ \neg P \land Q \]

### Final Answer:
The dual of \( P \implies Q \) is \( \neg P \land Q \).

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