Write 'T' for True and 'F' for False.
`~| (p implies q)=p ^^ ~| q`.

Answer

To determine the truth value of the statement `~| (p implies q) = p ^^ ~| q`, we will analyze both sides of the equation step by step.

### Step 1: Understand the components of the statement
The statement involves:
- `p implies q` which can be expressed as `p → q`.
- The negation of this implication, which is `~(p → q)`.
- `p ^^ ~| q`, where `~| q` represents the negation of `q`.

### Step 2: Rewrite `p implies q`
The implication `p → q` can be rewritten using logical equivalences:
\[ p \rightarrow q \equiv \neg p \lor q \]
This means that `p implies q` is equivalent to `not p or q`.

### Step 3: Negate the implication
Now we need to negate `p → q`:
\[ \neg (p \rightarrow q) \equiv \neg (\neg p \lor q) \]
Using De Morgan's Law, this can be rewritten as:
\[ \neg (\neg p \lor q) \equiv p \land \neg q \]
So, `~(p → q)` is equivalent to `p ∧ ¬q`.

### Step 4: Analyze the right side of the equation
Now we look at the right side of the equation, `p ^^ ~| q`:
- `~| q` is simply `¬q`.
- Therefore, `p ^^ ~| q` translates to `p ∧ ¬q`.

### Step 5: Compare both sides
Now we have:
- Left side: `~(p → q) ≡ p ∧ ¬q`
- Right side: `p ∧ ¬q`

Since both sides are equivalent:
\[ \neg (p \rightarrow q) = p \land \neg q \]

### Conclusion
Since both sides of the equation are equal, we conclude that the statement is true. Therefore, we write:
**Answer: T (True)**

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