Proving A -> (A -> B) Equivalent to (A -> B)

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In summary, the conversation discusses the formal proof of the logical equivalence between the statements "A implies (A implies B)" and "A implies B". The individual asks if they can assume A and use the modus ponens deduction method to prove the equivalence, and expresses concern about asserting a premise without justification. The other participant assures them that using a truth table is a valid and rigorous method of proof. The individual then presents a full proof using various logical rules, and asks for guidance on how to proceed with the steps involving asserting the premise Y.
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EvLer
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Is there a way to formally prove:
A -> (A -> B) equivalent to (A -> B)

do I just assume A and then use modus ponenes (by deduction method)? It just looks a bit odd asserting a premise out of the blue...
I checked the truth table for those, it seems to be true, however, truth table is not going to work as the answer, I need to do this formally.
Thanks in advance.

ps: sorry for trashing forum with my logic threads... from now on, i'll just put my questions (if i have any) here :redface:
 
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  • #2
I checked the truth table for those, it seems to be true
More than just "seems"! Any logical equivalence provable via truth table can be deduced with the rules of inference you're using, and vice versa.
 
  • #3
Proofs by exhaustion (like truth tables) are perfectly rigorous.
 
  • #4
well that is the thing... I check my "inference" steps with truth tables, however I need a rule or transformation of some sort and I can't see anything unless I assert the premise (A).
Ok, so here's the full thing:
prove Y -> W
given these premises:
1. Y -> Z'
2. X' -> Y
3. Y -> (X -> W)
4. Y -> Z
-----my proof----
5. X' -> Z' (2,1, hypoth. syllogism)
6. Z -> X (5, contraposition)
7. Y -> X (4,6, hypothet. syllogism)
8. (X' v W)' -> Y' (3, contraposition)
9. (X ^ W') -> Y' (DeMorgan's)
10. X -> (W' ->Y') (9, exportation)
11. X -> (Y ->W) (contraposition)

12. Y -> (Y -> W) (7,11, hypothet. syllogism)
13. Y (?)
14. Y -> W (?)
12-14 is where I need something or can I just assert Y?
 
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FAQ: Proving A -> (A -> B) Equivalent to (A -> B)

1. What does "Proving A -> (A -> B) Equivalent to (A -> B)" mean?

This statement is a mathematical or logical statement that is saying that if A implies B, and A also implies B implies something else, then the two statements are equivalent. In other words, they are saying the same thing and can be used interchangeably.

2. How can I prove that A -> (A -> B) is equivalent to (A -> B)?

There are several ways to prove this statement, but one way is to use a truth table. You can list out all possible combinations of A and B and show that the two statements have the same truth values for each combination. This will prove that they are equivalent.

3. Why is this statement important in science?

This statement is important in science because it is a fundamental concept in logic and reasoning. It helps us understand how different statements are related to each other and how we can use them to make logical conclusions. In science, we use this concept to build theories and make predictions based on evidence and logical reasoning.

4. Are there any real-life examples of this statement?

Yes, there are many real-life examples of this statement. One example is the concept of cause and effect. If we assume that A causes B, then we can also assume that if A happens, then B will happen. This is similar to saying A implies B. Another example is if you have a conditional statement like "If it rains, then the ground will be wet." This is equivalent to saying "If it doesn't rain, then the ground won't be wet."

5. Can this statement be proven wrong?

No, this statement cannot be proven wrong because it is a logical statement that is based on mathematical principles. As long as the rules of logic and reasoning are followed, this statement will always hold true. However, it is important to note that this statement is based on assumptions and can change if new evidence or information is presented.

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