The negation of `p implies q` is:

Options

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

Answer

To find the negation of the statement "p implies q" (denoted as \( p \implies q \)), we can follow these steps:

### Step 1: Understand the implication
The implication \( p \implies q \) can be expressed in terms of logical operations. It is equivalent to \( \neg p \lor q \) (not p or q).

### Step 2: Write the equivalence
Thus, we have:
\[ p \implies q \equiv \neg p \lor q \]

### Step 3: Negate the equivalence
To find the negation of \( p \implies q \), we need to negate the expression \( \neg p \lor q \):
\[ \neg(p \implies q) \equiv \neg(\neg p \lor q) \]

### Step 4: Apply De Morgan's Law
Using De Morgan's Law, we can convert the negation of a disjunction into a conjunction:
\[ \neg(\neg p \lor q) \equiv \neg(\neg p) \land \neg(q) \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ \neg(\neg p) \land \neg(q) \equiv p \land \neg q \]

### Conclusion
Thus, the negation of \( p \implies q \) is:
\[ \neg(p \implies q) \equiv p \land \neg q \]

### Final Answer
The negation of \( p \implies q \) is \( p \land \neg q \).

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