Write 'T' for True and 'F' for False.
`~| (p implies q) vv r` is a dual of `~| (p implies q) ^^ r`.

Answer

To determine whether the statement `~| (p implies q) vv r` is a dual of `~| (p implies q) ^^ r`, we will analyze both expressions step by step.

### Step 1: Understand the expressions
The first expression is `~| (p implies q) vv r`, and the second expression is `~| (p implies q) ^^ r`. Here, `~|` denotes negation, `vv` denotes disjunction (logical OR), and `^^` denotes conjunction (logical AND).

### Step 2: Rewrite the implications
The implication `p implies q` can be rewritten using logical operators:
- `p implies q` is equivalent to `~p vv q`.

### Step 3: Substitute the implication
Now, substitute `p implies q` in both expressions:
1. For the first expression:
- `~| (p implies q) vv r` becomes `~| (~p vv q) vv r`.
2. For the second expression:
- `~| (p implies q) ^^ r` becomes `~| (~p vv q) ^^ r`.

### Step 4: Apply negation
Now, we need to apply the negation to both expressions:
1. The negation of `~p vv q` is `p ^^ ~q` (using De Morgan's laws).
2. Therefore:
- The first expression becomes `p ^^ ~q vv r`.
- The second expression becomes `p ^^ ~q ^^ r`.

### Step 5: Identify the dual
According to the rules of duality:
- Disjunction (vv) is converted to conjunction (^^) and vice versa.
- Therefore, the dual of `p ^^ ~q vv r` is `p ^^ ~q ^^ r`.

### Step 6: Compare the two expressions
Now we compare the two expressions:
- The first expression after transformation is `p ^^ ~q vv r`.
- The dual of this expression is `p ^^ ~q ^^ r`.

Since we see that the first expression is a disjunction and the second is a conjunction, they are not equivalent.

### Conclusion
Thus, the statement `~| (p implies q) vv r` is **not** a dual of `~| (p implies q) ^^ r`. Therefore, the answer is **F** (False).

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