I'm noob practicing with discrete math problems, and not sure if the solution ArsDigita provided for this one is correct or not:
Prove that a → b is equivalent to ¬b → ¬a using algebraic identities.
Since the materials were provided 10 years ago, I'm not sure if there could be any correction and I haven't found any equivalent discussion. Therefore I'm here to seek for help.
The answer offered is like below:
a → b ⇔ ¬a ∧ b
⇔ b ∧ ¬a
⇔ (¬¬b) ∧ ¬a
⇔ ¬b → ¬a
I think it should be a → b ⇔ ¬a v b? Could you help to confirm this?
I'm really confused and would also have your advises to see if I should keep following ArsDigita's materials. I enjoy the professor's video lectures very much, but if there's error it would be very inefficient to keep on.
Please let me know. Thanks in advance for your help.
$$A\to B\Longleftrightarrow\neg A\lor B\Longleftrightarrow \neg A\lor\neg\neg B\Longleftrightarrow \neg(\neg B)\lor(\neg A)\Longleftrightarrow \neg B\to\neg A$$