1. Is \( A \vee A \) not a wff?2. Is \( A + A \) a wff?3. Is \( A \& \sim A \) a wff?4. Is \( \sim (\sim Z \supset (Z \vee Z)) \) a wff?5. Is \( (A \supset B) \& (C \equiv D) \vee ((E \vee F) G) \) a wff?6. Is \( A \sim \supset (E \vee D) \) a wff?7. Is \( A \supset \sim A \) a wff?8. Is \( \sim A \supset (A \vee A) \) a wff?9. Is \( A \sim Z \supset A \vee A \) a wff?10. Is \( A \sim A \supset \sim B \) a wff?

In logic, a well-formed formula (wff) refers to a syntactically correct expression constructed according to the formal rules of the logical system. Let's analyze each of your expressions in terms of their well-formedness.

1. **\( A \vee A \)**:  
   Yes, this is a **wff**.  
   It follows the correct syntax for propositional logic where \( \vee \) represents disjunction (OR). The same propositional variable appearing twice doesn't affect the formula's well-formedness.

2. **\( A + A \)**:  
   No, this is **not a wff**.  
   The symbol \( + \) is not a standard logical operator in classical propositional logic. It is more commonly used in arithmetic or boolean algebra.

3. **\( A \& \sim A \)**:  
   Yes, this is a **wff**.  
   \( \& \) represents conjunction (AND), and \( \sim A \) is a valid negation of \( A \). The formula \( A \& \sim A \) is well-formed.

4. **\( \sim (\sim Z \supset (Z \vee Z)) \)**:  
   Yes, this is a **wff**.  
   Negation \( \sim \), implication \( \supset \), and disjunction \( \vee \) are all valid logical operators. This formula is properly constructed.

5. **\( (A \supset B) \& (C \equiv D) \vee ((E \vee F) G) \)**:  
   No, this is **not a wff**.  
   The subexpression \( (E \vee F) G \) is not well-formed because the formula should contain logical operators between propositional variables. The juxtaposition of \( (E \vee F) \) and \( G \) is syntactically incorrect.

6. **\( A \sim \supset (E \vee D) \)**:  
   No, this is **not a wff**.  
   \( A \sim \supset \) is not a valid combination. The negation operator \( \sim \) should precede a proposition or formula, but here it is placed between two components in an incorrect manner.

7. **\( A \supset \sim A \)**:  
   Yes, this is a **wff**.  
   Implication \( \supset \) and negation \( \sim A \) are used correctly. This formula represents "if \( A \) then not \( A \)."

8. **\( \sim A \supset (A \vee A) \)**:  
   Yes, this is a **wff**.  
   This formula is well-formed. It consists of a negation \( \sim A \), implication \( \supset \), and a valid disjunction \( (A \vee A) \).

9. **\( A \sim Z \supset A \vee A \)**:  
   No, this is **not a wff**.  
   The subexpression \( A \sim Z \) is not valid, as \( \sim \) should apply to a single propositional variable or formula, but here it is placed between \( A \) and \( Z \) incorrectly.

10. **\( A \sim A \supset \sim B \)**:  
   No, this is **not a wff**.  
   The subexpression \( A \sim A \) is not valid. As mentioned earlier, \( \sim \) should apply to a single formula, not between two propositional variables.

In summary, the well-formed formulas (wffs) are:  
- \( A \vee A \)  
- \( A \& \sim A \)  
- \( \sim (\sim Z \supset (Z \vee Z)) \)  
- \( A \supset \sim A \)  
- \( \sim A \supset (A \vee A) \)

Would you like more details or have any other questions?

### 5 Related Questions:
1. What are the formal rules for constructing well-formed formulas in propositional logic?
2. Can you explain the difference between logical operators like \( \supset \), \( \vee \), and \( \& \)?
3. What are some examples of non-wffs in propositional logic and why they fail?
4. How does propositional logic differ from first-order logic in terms of wff construction?
5. What role does syntax play in ensuring a formula is a wff?

### Tip:
Always ensure that logical operators are applied correctly and that every part of a formula follows the syntactic rules of the logic system in use.

1. Is ( A ∨ A ) not a wff? 2. Is ( A + A ) a wff? 3. Is ( A & ~ A ) a wff? 4. Is ( ~ (~ Z → (Z ∨ Z)) ) a wff? 5. Is ( (A → B) & (C ≡ D) ∨ ((E ∨ F) G) ) a wff? 6. Is ( A ~ → (E ∨ D) ) a wff? 7. Is ( A → ~ A ) a wff? 8. Is ( ~ A → (A ∨ A) ) a wff? 9. Is ( A ~ Z → A ∨ A ) a wff? 10. Is ( A ~ A → ~ B ) a wff?

Analyze various logical expressions to determine if they are well-formed formulas (WFF) in propositional logic. This solution covers concepts like disjunction, conjunction, implication, and negation, suitable for college-level logic studies.

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