ONT Re: Set Theory
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Note 8
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| Elementary Set Theory
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| Elementary Algebra of Classes (cont.)
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| 12. Theorem. (De Morgan).
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| ~(x |_| y) = (~x) |^| (~y)
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| and
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| ~(x |^| y) = (~x) |_| (~y).
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| Proof. Only the first of the two statements will be proved.
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| For each z, we have z in ~(x |_| y) iff z is a set
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| and it is false that z in (x |_| y), because of the
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| classification axiom and the definition 10 of complement.
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| Using theorem 4, z in x |_| y iff z in x or z in y.
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| Consequently, z in ~(x |_| y) iff z is a set and
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| z ~in x and z ~in y, that is, iff z in ~x and z in ~y.
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| Using 4 again, z in ~(x |_| y) iff z in (~x) |^| (~y).
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| Hence ~(x |_| y) = (~x) |^| (~y) because of the
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| axiom of extent. þ
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| 13. Definition. x ~ y = x |^| (~y).
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| The class x ~ y is the 'difference' of x and y,
| or the 'complement' of y relative to x.
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| 14. Theorem. x |^| (y ~ z) = (x |^| y) ~ z.
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| The proposition "x |_| (y ~ z) = (x |_| y) ~ z" is unlikely,
| although at this stage it is impossible to exhibit a counter example.
| To be a little more precise, the negation of the proposition cannot be
| proved on the basis of the axioms so far assumed; it is possible to
| make a model for this initial part of the system such that x ~in y
| for each x and each y (there are no sets). The negation of the
| proposition can be proved on the basis of axioms which will
| presently be assumed.
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| JLK, Gen Top, pages 254-255.
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| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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