ONT Re: Set Theory
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Note 9
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| Elementary Set Theory
|
| Elementary Algebra of Classes (cont.)
|
| 15. Definition. 0 = {x : x =/= x}.
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| The class 0 is the 'void class', or 'zero'.
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| 16. Theorem. x ~in 0.
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| 17. Theorem. 0 |_| x = x
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| and 0 |^| x = 0.
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| 18. Definition. $U$ = {x : x = x}.
|
| The class $U$ is the 'universe'.
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| 19. Theorem. x in $U$ if and only if x is a set.
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| 20. Theorem. x |_| $U$ = $U$
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| and x |^| $U$ = x.
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| 21. Theorem. ~0 = $U$
|
| and ~$U$ = 0.
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| 22. Definition. †
|
| |^| x = {z : for each y, if y in x, then z in y}.
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| 23. Definition.
|
| |_| x = {z : for some y, z in y and y in x}.
|
| The class |^| x is the 'intersection' of the members of x.
| Note that the members of |^| x are members of members of x
| and may or may not belong to x.
|
| The class |_| x is the 'union' of the members of x.
|
| Observe that a set z belongs to |^| x (or to |_| x) iff
| z belongs to every (respectively, to some) member of x.
|
| 24. Theorem. |^| 0 = $U$
|
| and |_| 0 = 0.
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| Proof. z in |^| 0 iff z is a set and z belongs to each member of 0.
|
| Since (theorem 16) there is no member of 0, z in |^| 0 iff
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| z is a set, and by 19 and the axiom of extent |^| 0 = $U$.
|
| The second assertion is also easy to prove. þ
|
| † A bound variable notation for the intersection
| of the members of a family is not needed in this
| appendix, and consequently a notation is adopted
| which is simpler than that used in the rest of
| the book.
|
| JLK, Gen Top, pages 255-256.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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