A nanodomain is an object studied in the field of mathematical stereotopodynamics.  A nanodomain may be holograded, or not, and it may be pre-associative, or not.  You are given a list of statements concerning a nanodomain X; some of these statements are logically equivalent, that is, they are just different ways of saying the same thing.      Group the statements into logically equivalent sets and enter your answer below as a list of sets separated by commas.  Syntax advice:  For example, if you think that statements 1,2,3 are logically equivalent; and statements 4,5,6,7 are logically equivalent (but different from 1,2,3); and statement 8 is different from all the others; then your answer should be  {1,2,3}, {4,5,6,7}, {8}  The order of your sets, and the order of the elements in each set, are not important.     (1)  X is holograded and X is not pre-associative  (2)  X is pre-associative if X is holograded  (3)  if X is holograded, then X is pre-associative  (4)  X is holograded only if X is pre-associative  (5)  if X is not pre-associative, then X is holograded  (6)  X is holograded, or X is not pre-associative  (7)  if X is pre-associative, then X is holograded  (8)  X is holograded, or X is pre-associative. Use the fact that p->q is equivalent to not p or q

To solve this problem, we need to group the logically equivalent statements. Let's start by analyzing each statement:

1. **Statement (1):** X is holograded and X is not pre-associative.
2. **Statement (2):** X is pre-associative if X is holograded. This can be rewritten as: If X is holograded, then X is pre-associative. \( p \rightarrow q \).
3. **Statement (3):** If X is holograded, then X is pre-associative. This is the same as statement (2).
4. **Statement (4):** X is holograded only if X is pre-associative. This is the same as: If X is holograded, then X is pre-associative. \( p \rightarrow q \), which is equivalent to statements (2) and (3).
5. **Statement (5):** If X is not pre-associative, then X is holograded. \( \neg q \rightarrow p \). By the contrapositive, this is equivalent to \( \neg p \rightarrow \neg q \), which means that if X is not holograded, then X is pre-associative. This is equivalent to the negation of statement (4), \( \neg (p \rightarrow q) \), which is not equivalent to statements (2), (3), or (4).
6. **Statement (6):** X is holograded, or X is not pre-associative. This can be written as \( p \lor \neg q \). By logical equivalence, \( p \lor \neg q \) is equivalent to \( \neg(\neg p \land q) \), which means it is the negation of the statement "X is not holograded and X is pre-associative." This statement is equivalent to statement (1), but not equivalent to others.
7. **Statement (7):** If X is pre-associative, then X is holograded. \( q \rightarrow p \). This is the converse of statements (2), (3), and (4). This is different from the previous statements.
8. **Statement (8):** X is holograded, or X is pre-associative. \( p \lor q \). This is a different statement from the others.

**Group the statements:**

- **{2, 3, 4}:** These all express the idea that if X is holograded, then X is pre-associative.
- **{1, 6}:** These both imply that X is holograded and X is not pre-associative.
- **{5, 7}:** These express conditional equivalences that are logically converse to each other.
- **{8}:** This statement stands alone, expressing that at least one of the properties (holograded or pre-associative) must hold.

So, the correct grouping of the statements is:

**{2, 3, 4}, {1, 6}, {5, 7}, {8}**

Would you like more details or have any other questions? Here are five related questions to explore:

1. What does it mean for a statement to be logically equivalent to another?
2. How can you use the contrapositive to determine logical equivalence?
3. What are some common logical equivalences in propositional logic?
4. Can two statements with different syntaxes be logically equivalent? How?
5. How does De Morgan's law help in simplifying logical statements?

**Tip:** When analyzing logical statements, always check if you can apply common logical equivalences like contrapositive, De Morgan's laws, or implications to simplify the expressions.

Learn how to group logically equivalent statements in propositional logic. Understand concepts such as contrapositive and De Morgan's laws to identify equivalent logical expressions. Suitable for advanced mathematics students or those studying propositional logic.

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