https://file.aiquickdraw.com/mathbot/file/1725797675256ntj9yuo6.jpg

The image presents a logical argument in the form of a proof:

1. **Premise:** \( A \to B \)  
2. **Conclusion:** \( (A \to B) \lor \neg C \)  

### Reasoning

To derive the conclusion, \( (A \to B) \lor \neg C \), from the premise \( A \to B \):

- The premise \( A \to B \) asserts that if \( A \) is true, then \( B \) must also be true.
- The conclusion \( (A \to B) \lor \neg C \) is a disjunction that states either "if A then B" is true or "not C" is true.

Since the premise \( A \to B \) is already assumed to be true, the disjunction \( (A \to B) \lor \neg C \) will always be true regardless of whether \( C \) is true or false.

### Conclusion

The argument is valid because the conclusion \( (A \to B) \lor \neg C \) logically follows from the premise \( A \to B \). In other words, adding a disjunction with a true statement does not affect the truth value of the logical expression.

Would you like a more detailed explanation or to ask any questions?

### Related Questions

1. What are some common rules of inference in propositional logic?
2. How does the law of disjunction work in logical proofs?
3. What is the difference between a tautology and a contradiction?
4. How do truth tables help verify the validity of logical arguments?
5. What are De Morgan's laws, and how are they applied in logical proofs?

#### Tip:
When working with logical proofs, it helps to break down each step and apply fundamental logical laws to see how one statement leads to another.

Given A → B, prove that (A → B) ∨ ¬C follows logically.

This proof demonstrates how the disjunction (A → B) ∨ ¬C logically follows from the premise A → B. It covers essential concepts of propositional logic such as implications, disjunction, and tautology, suitable for students studying discrete math or logic at the undergraduate level.

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