- #36
Dmobb Jr.
- 39
- 0
First define 0: 0 = {}
Define 1: 1={{}}
Define 2: 2={{},{{}}}
Define =: A = B iff (for all x (x in A iff x in B)
In other words, two sets are equal if they have the same elements.
Define S(): S(x) = xU{x}
Define +:
1) A + 0 = A
2) A + S(B) = S(A) + B
Now for the proof:
Clearly S(0) = S({}) = {} U {{}} = {{}} = 1
and S(1) = S({{}}) = {{}}U{{{}}} = {{},{{}}} = 2
So 1 = S(0) and S(1) = 2
1 + 1 = 1 + S(0) = S(1) + 0 = S(1) = 2
Then again that is just as valid as defining 2 as 1+1.
Define 1: 1={{}}
Define 2: 2={{},{{}}}
Define =: A = B iff (for all x (x in A iff x in B)
In other words, two sets are equal if they have the same elements.
Define S(): S(x) = xU{x}
Define +:
1) A + 0 = A
2) A + S(B) = S(A) + B
Now for the proof:
Clearly S(0) = S({}) = {} U {{}} = {{}} = 1
and S(1) = S({{}}) = {{}}U{{{}}} = {{},{{}}} = 2
So 1 = S(0) and S(1) = 2
1 + 1 = 1 + S(0) = S(1) + 0 = S(1) = 2
Then again that is just as valid as defining 2 as 1+1.