What represents the truth set of the Negation of an open sentence $P_{x}$?

Question

Accepted Answer

The correct answer to the question "What represents the truth set of the Negation of an open sentence $ P_{x} $?" is Option 3: The complement of the truth set of $ P_{x} $.

### Explanation for Option 3:

The truth set of an open sentence $ P_{x} $ is the set of all elements in the universal set $ U $ for which the sentence $ P_{x} $ is true. When we talk about the negation of an open sentence, denoted as $ \neg P_{x} $, we are referring to the set of elements for which $ P_{x} $ is false. This set is precisely the complement of the truth set of $ P_{x} $ within the universal set $ U $.

In set theory, the complement of a set $ A $ with respect to a universal set $ U $ is the set of elements in $ U $ that are not in $ A $. Therefore, the truth set of $ \neg P_{x} $ is the complement of the truth set of $ P_{x} $.

### Explanation for why the other options are incorrect:

1. **Option 1: The union of the universal set $ U $ and $ P_{x} $:**
   - The union of the universal set $ U $ and any subset $ P_{x} $ is simply the universal set $ U $ itself, since $ U $ already contains all elements. This does not represent the truth set of the negation of $ P_{x} $, as it includes elements where $ P_{x} $ is true and false.

2. **Option 2: The intersection of $ P_{x} $ and $ U $:**
   - The intersection of $ P_{x} $ and the universal set $ U $ is just $ P_{x} $ itself, as $ P_{x} $ is a subset of $ U $. This does not represent the truth set of the negation of $ P_{x} $, as it only includes elements where $ P_{x} $ is true.

3. **Option 4: The truth set of $ P_{x} $ itself:**
   - The truth set of $ P_{x} $ consists of elements for which $ P_{x} $ is true. The negation of $ P_{x} $ would include elements where $ P_{x} $ is false, which is the opposite of the truth set of $ P_{x} $. Therefore, this option does not represent the truth set of the negation of $ P_{x} $.

In conclusion, Option 3 is the correct answer because the truth set of the negation of an open sentence $ P_{x} $ is indeed the complement of the truth set of $ P_{x} $ within the universal set $ U $.