@Stupidquestioninc I have to say that your instructor is not teaching you properly. A proof does not involve English 'reasoning'. In particular, never use "doesn't imply" if you want to do proper logical reasoning.
If A = false and B = true, ( A ⇒ B ) = true by definition of "⇒". There is absolutely nothing to prove.
On the other hand, you also have to know how to capture that in a formal proof in a deductive system for PL. That is, you must be able to write a formal proof of ( ¬A , B ⊢ A ⇒ B ), namely a proof that from ¬A and B deduces ( A ⇒ B ).
Equivalently, you must be able to write a formal proof of ( ¬A∧B ⇒ ( A ⇒ B ) ).
I think I told you to learn this system before. If you want to learn logic properly, you must learn to write proofs in a formal system like that one. A proof of ( ¬A∧B ⇒ ( A ⇒ B ) ) would look like:
If ¬A∧B:
¬A.
B.
If A:
...
B.
A ⇒ B.
¬A∧B ⇒ ( A ⇒ B ).
In this case, the proof is so trivial that there is nothing to fill in at "..."...
I want you to instead prove ( ¬A ⇒ ( A ⇒ B ) ), which is slightly less trivial.
@Stupidquestioninc ( X ⊢ Y ) is a conventional notation meaning "from X you can prove Y".
@Stupidquestioninc Yes the meaning of "⇒" is defined by its truth-table. It takes 2 inputs, each of which is a boolean, and produces an output that is also a boolean given by the truth-table. Since there are 4 possible input combinations, the truth-table has 4 rows to specify the output in each case.
@Stupidquestioninc Thank you, I put in a lot of effort into my posts, especially those that I use to teach others. =)
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@Stupidquestioninc I will come back to your real analysis question later. Busy now.
