Help with some set theory questions

AI Thread Summary
Assistance is requested for several set theory proofs, specifically regarding operations involving intersections and symmetric differences. The user has made attempts at the first proof but is seeking further guidance. Participants suggest reverse engineering the proofs to better understand the relationships between the sets involved. The discussion emphasizes collaborative problem-solving to clarify set theory concepts. Engaging with the community can enhance comprehension of these mathematical principles.
Beno
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Greetings all. I was wondering if someone could give me some asistance on a few simple set theory proofs.

1) An(B\C)=(AnB)\(AnC)
2) A+(B+C)=(A+B)+C where + = symmetric difference
3) If A+B = A+C then B=C
and
4) An(B+C) = (AnB)+(AnC)
 
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Welcome to PF!

Hi Beno! Greetings and welcome to PF! :wink:

Show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 
With number 1
An(B\C)=(AnB)\(AnC)
I tried the following
An(B\C) = An(BnC') =(AnB)n(AnC') but this was as far as it seemed to go.

As for the others I think I've worked them out since posting
 
Beno said:
With number 1
An(B\C)=(AnB)\(AnC)
I tried the following
An(B\C) = An(BnC') =(AnB)n(AnC') but this was as far as it seemed to go.

ok, let's do some reverse engineering :wink: …

the last step but one is going to be (AnB)n(AnC)' …

work backwards from there. :smile:​
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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