Are (A then B) then C and (A and B) then C Equivalent?

[Gamecockgirl]

I need help proving that (A then B) then C and (A and B) then C are equivalent. Can anyone help?

 

[CompuChip]

There are several ways to prove such a thing. The easiest is using truth tables.
First of all, do you understand intuitively why it is true?

 

[Gamecockgirl]

I know why they are true and I have done the truth table however our teacher wants us to do it by proofs and I can't seem to make sense of the proof. I appreciate any guidance you can offer.

 

[Borek]

Please elaborate what "by proofs" means. Using truth table is a way of proving that expression is valid.

 

[Gamecockgirl]

By proof I mean using sentintial derivations where you have to use v introduction or elimination. & introduction or & elimination.

 

[CompuChip]

So first let's consider the direct implication and work backwards. Suppose you want to prove X => Y, where X is (A => B) => C and Y is (A & B) => C. What would be your final step and which assumptions and conclusions would you need?

 

[Gamecockgirl]

To prove ( A then B) then C your assumption would be (A and B) then C with your conclusion being (A then B) then C. To prove (A and B) then C your assumption would be (A then B) then C and your conclusion would be (A and B) then C. But I don't know anymore than that. That is the only thing I have been able to figure out so far. I think there might be some sub proofs and some more assumptions but I don't know what.

 

[CompuChip]

You just told me that: "to prove X, your assumption would be X with your conclusion being X. To prove Y your assumption would be Y with your conclusion being Y".

Maybe I was a bit too vague, so let me reformulate my question. You want to prove something of the form X => Y, so your final step will likely be =>I (implication introduction). So what do you start with and what do you need to prove, in order to be able to make this step?