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Hello, I have some problems proving the following problem: "AvB |- BvA". I would like to prove this by using natural deduction only. (I know it is possible proving it using truth tables).

Syntaxis

To avoid confusion I will use the following syntaxis to show my problem.

- Capital letters are used for variables.

- Instead of the mathematically symbols like "v" which means OR, I will use OR (all capital to substitute the mathematical symbols (few examples: (A AND B), (NOT(A))).

- I use something that I would like to call a diagram proof and hopefully you can explain how to do it using my familiar proof technique. I will demonstrate some below.

- I use |- to represent the mathematical symbol. It means: "When a conclusion C is derived from a set of assumptions {A1, A2, A3, ... , An} we write {A1, A2, A3, ... , An} |- C".

- When discharging a assumption I prefer to use [A] arround the proposition.

Diagrams

Note: I will use dots to represent spacings.

Note: I will use as much brackets as I can to make everything clear.

{A AND B} |- (B AND A)

(A.AND.B)..........(A.AND.B.)...........(Assumption)

----------..........---------

....B........................A................ (by AND elimination)

--------------------------

.............B.AND.A.........................(by AND introduction)

(please also add the rules used when explaining to me)

Problem Statement

Prove using natural deduction: (A OR B) |- (B OR A)

The attempt at a solution

Using proof by contradiction:

.....................................................................[NOT(B.OR.A)]......(Assumption)

..................................................................-------------------

AvB.........[NOT(A)].................(Assumption)........NOT(B).AND.NOT(A).(Rewrite)

-----------------..........................................--------------------

.B.......................................................................NOT(B)..............(AND elimination)

-------------------------------------------------------------------

...................................(B.AND.NOT(B))..........................................(AND introduction)

--------------------------------------------------------------------

.......................................FALSE...................................................

--------------------------------------------------------------------

......................................(B OR A).......................................(Proof by contradiction)

Is this correct? Because I've the feeling I'm missing something.

Questions

A few questions about natural deduction itself.

- Can I use every assumption as long I discharge them later in the proof? (except for those that are already given in the excersize?

Thanks for explaining everything.

Greetings TKM

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# Homework Help: Natural Deduction - AvB |- BvA

Can you offer guidance or do you also need help?

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