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For A, B We say that A is a subset of B, or A is contained in B if and only if for any x, if x is in A then x is in B. We say that A is a proper subset of B, or A is properly contained in B if and only if "A is contained in B but A is not B". We write down an axiom afterward.
(Power set axiom) For any A, these is an X such that for any Y, "Y is in X if and only if Y is a subset of A".
With all axioms above, we can further define the substraction of A by B, A\B, the power set of A, Pow(A), or abbr(??) P(A).
There is a theorem saying that, there is a UNIQUE set C such that the following (i) or (ii) is true:
(i) for any x, x is in C if and only if P(x).
(ii) C is the empty set and there is no such set B such that for any t, t is in B if and only if P(t).
The proof requires linking between intuitive thinking and mathematical writing. We divid it into cases: (a) There is a set K such that for any x, x is in K if and only if P(x), and (b), there is no such K such that for any x, x is in K if and only if P(x). Then choose a C.
This theorem gives the permission of giveing a definition called Definition Schema, which is nowaday the statement expression(???).
The set C for (i) or (ii) above to hold is called "the set of those x such that P(x)", denoted by {x: P(x)}, that is, C={x: P(x)} if and only if "for any x, x is in C if and only if P(x)", or, "C is the empty set and there is no such set B such that for any t, t is in B if and only if P(t)".
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