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

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Discussion Overview

The discussion revolves around the equivalence of the logical expressions (A then B) then C and (A and B) then C. Participants are seeking methods to prove this equivalence, specifically through formal proofs rather than truth tables.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests help in proving the equivalence of the two expressions.
  • Another suggests that truth tables can demonstrate the equivalence but questions whether the original poster understands the intuition behind it.
  • A participant indicates they have completed a truth table but are required to provide a proof using formal derivations.
  • Clarification is sought on what is meant by "by proofs," with a distinction made between truth tables and formal proofs.
  • One participant explains that they need to use rules like introduction and elimination in their proofs.
  • A suggestion is made to consider the implications directly and work backwards to establish the proof structure.
  • Another participant attempts to outline the assumptions and conclusions needed for the proof but expresses uncertainty about the process and potential sub-proofs.
  • There is a critique of the previous reasoning, emphasizing the need for a clear starting point and what must be proven to apply implication introduction.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the best method to prove the equivalence, and there are multiple competing views on how to approach the proof.

Contextual Notes

Participants express uncertainty regarding the assumptions and steps needed in formal proofs, indicating that the discussion may be limited by varying interpretations of proof techniques.

Gamecockgirl
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I need help proving that (A then B) then C and (A and B) then C are equivalent. Can anyone help?
 
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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?
 
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.
 
Last edited:
Please elaborate what "by proofs" means. Using truth table is a way of proving that expression is valid.
 
By proof I mean using sentintial derivations where you have to use v introduction or elimination. & introduction or & elimination.
 
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?
 
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.
 
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?
 

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