ONT Re: Set Theory
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Note 13
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| Elementary Set Theory
|
| Existence of Sets (cont.)
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| 40. Definition. {x} = {z : if x in $U$, then z = x}.
|
| 'Singleton' x is {x}.
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| This definition is an example of a technical device which is very convenient.
| If x is a set, {x} is a class whose only member is x. However, if x is not
| a set, then {x} = $U$ (these statements are theorems 41 and 43). Actually,
| the primary interest is in the case where x is a set, and for this case
| the same result is given by the more natural definition {z : z = x}.
| However, it simplifies statements of results greatly if computations
| are arranged so that $U$ is the result of applying the computation
| outside its pertinent domain.
|
| 41. Theorem.
|
| If x is a set,
| then, for each y,
| y in {x} iff y = x.
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| 42. Theorem.
|
| If x is a set, then {x} is a set.
|
| Proof. If x is a set, then {x} c 2^x, and 2^x is a set. þ
|
| 43. Theorem.
|
| {x} = $U$ if and only if x is not a set.
|
| Proof. If x is a set, then {x} is a set
| and consequently is not equal to $U$.
| If x is not a set, then x ~in $U$ and
| and {x} = $U$ by the definition. þ
|
| 44. Theorem.
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| If x is a set,
|
| then |^| {x} = x
|
| and |_| {x} = x.
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| If x is not a set,
|
| then |^| {x} = 0
|
| and |_| {x} = $U$.
|
| Proof. Use 34 and 41. þ
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| IV. Axiom of Union.
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| If x is a set and y is a set so is x |_| y.
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| 45. Definition. {x y} = {x} |_| {y}.
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| The class {x y} is an 'unordered pair'.
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| 46. Theorem.
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| If x is a set and y is a set,
|
| then {x y} is a set and z in {x y} iff z = x or z = y.
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| {x y} = $U$ if and only if x is not a set or y is not a set.
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| 47. Theorem.
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| If x and y are sets,
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| then |^| {x y} = x |^| y
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| and |_| {x y} = x |_| y.
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| If either x or y is not a set,
|
| then |^| {x y} = 0
|
| and |_| {x y} = $U$.
|
| JLK, Gen Top, page 258.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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