II. Notes on Data Structuring * - Cornell University

12.5 POWERSETS
NOTES ON DATA STRUCTURING 171
The axioms given below for sets apply only to finite sets of hierarchically
ordered type. It is therefore possible to avoid the paradoxes which endanger
axiomatigations of more powerful versions of set theory.
The general form of a powerset definition is:
type T = powerset To,
where To is the base type.
let a, b, be values of type To.
(1) T( )isaT
(2) If x is a T and a is a To then
x v T(a) isaT
(3) The only members of T are as specified in (1) and (2).
(4) --lain T( )
(5) a in (y v T(a))
(6) a ~ b = (a in (x v T(b)) ~ a in x)
(7) T( )=x
(8) (y v T(a)) = x=- (y = x&ainx)
(9) x=y~-(x=y)&(y=x)
(10) xv T()=x
(1 l) x v (y v T(a)) = (x v T(a)) v y
(12) x^r()=T()
(13) x ^ T(a) = if a in x then T(a) else T( )
(14) x ^ (y v T(a))= (x A y) V (x A T(a))
(15) T( )-x= T( )
(16) T(a) - x = if a in x then T( ) else T(a)
(17) (x v T(a))- y = (x- y) v (T(a)- y)
(18) size(T()) = 0
(19) size (x v T(a)) = if a in x then size (x) else succ (size (x))
The following apply if the domain type To is ordered:
(20) min (T(a))= T(a)
(21) x ~ T( ) = min (x v T(a)) = if a < min (x) then a else min (x)
Note: min (T()) is not defined
(22) xdown0=xup0=x
: E