Use De Morgan's Laws to write the negation of each statement: The sun is shining and it is raining. The sun is shining and it is not raining. The sun is shining or it is not raining. The sun is not shining and it is raining. The sun is not shining or it is raining. None of these. The dog is not sleeping and the dog is snoring. The dog is not sleeping or the dog is snoring. The dog is not sleeping or the dog is not snoring. The dog is sleeping or the dog is not snoring. The dog is sleeping and the dog is not snoring. None of these.

To apply De Morgan's Laws, we need to understand how to negate statements involving "and" (conjunction) and "or" (disjunction). De Morgan's Laws state: 1. The negation of a conjunction: \(\neg (A \land B) \equiv \neg A \lor \neg B\) 2. The negation of a disjunction: \(\neg (A \lor B) \equiv \neg A \land \neg B\) Now, let's apply these laws to each statement: 1. **The sun is shining and it is raining.** - Negation: The sun is not shining or it is not raining. - Symbolically: \(\neg (S \land R) \equiv \neg S \lor \neg R\) 2. **The sun is shining and it is not raining.** - Negation: The sun is not shining or it is raining. - Symbolically: \(\neg (S \land \neg R) \equiv \neg S \lor R\) 3. **The sun is shining or it is not raining.** - Negation: The sun is not shining and it is raining. - Symbolically: \(\neg (S \lor \neg R) \equiv \neg S \land R\) 4. **The sun is not shining and it is raining.** - Negation: The sun is shining or it is not raining. - Symbolically: \(\neg (\neg S \land R) \equiv S \lor \neg R\) 5. **The sun is not shining or it is raining.** - Negation: The sun is shining and it is not raining. - Symbolically: \(\neg (\neg S \lor R) \equiv S \land \neg R\) 6. **None of these.** - This statement does not require negation as it is a standalone option. 7. **The dog is not sleeping and the dog is snoring.** - Negation: The dog is sleeping or the dog is not snoring. - Symbolically: \(\neg (\neg D \land S) \equiv D \lor \neg S\) 8. **The dog is not sleeping or the dog is snoring.** - Negation: The dog is sleeping and the dog is not snoring. - Symbolically: \(\neg (\neg D \lor S) \equiv D \land \neg S\) 9. **The dog is not sleeping or the dog is not snoring.** - Negation: The dog is sleeping and the dog is snoring. - Symbolically: \(\neg (\neg D \lor \neg S) \equiv D \land S\) 10. **The dog is sleeping or the dog is not snoring.** - Negation: The dog is not sleeping and the dog is snoring. - Symbolically: \(\neg (D \lor \neg S) \equiv \neg D \land S\) 11. **The dog is sleeping and the dog is not snoring.** - Negation: The dog is not sleeping or the dog is snoring. - Symbolically: \(\neg (D \land \neg S) \equiv \neg D \lor S\) 12. **None of these.** - This statement does not require negation as it is a standalone option. In summary, the negations of the statements are as follows: 1. The sun is not shining or it is not raining. 2. The sun is not shining or it is raining. 3. The sun is not shining and it is raining. 4. The sun is shining or it is not raining. 5. The sun is shining and it is not raining. 6. None of these. 7. The dog is sleeping or the dog is not snoring. 8. The dog is sleeping and the dog is not snoring. 9. The dog is sleeping and the dog is snoring. 10. The dog is not sleeping and the dog is snoring. 11. The dog is not sleeping or the dog is snoring. 12. None of these.