ONT Re: Set Theory
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Note 14
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| Elementary Set Theory
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| Ordered Pairs: Relations
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| This section is devoted to the properties of ordered pairs and relations.
| The crucial property for ordered pairs is theorem 55: if x and y are sets,
| then (x, y) = (u, v) iff x = u and y = v.
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| 48. Definition. (x, y) = {{x} {x y}}.
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| The class (x, y) is an 'ordered pair'.
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| 49. Theorem.
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| (x, y) is a set if and only if x is a set and y is a set.
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| If (x, y) is not a set, then (x, y) = $U$.
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| 50. Theorem.
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| If x and y are sets, then:
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| |_| (x, y) = {x y}
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| |^| (x, y) = {x}
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| |_| |_| (x, y) = x |_| y
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| |_| |^| (x, y) = x
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| |^| |_| (x, y) = x |^| y
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| |^| |^| (x, y) = x
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| If either x or y is not a set, then:
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| |_| |_| (x, y) = $U$
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| |_| |^| (x, y) = 0
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| |^| |_| (x, y) = 0
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| |^| |^| (x, y) = $U$
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| 51. Definition. 1st coord z = |^| |^| z.
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| 52. Definition. 2nd coord z = (|^| |_| z) |_| ((|_| |_| z) ~ |_| |^| z).
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| These definitions will be used, with one exception,
| only in the case where z is an ordered pair.
| The 'first coordinate' of z is 1st coord z.
| The 'second coordinate' of z is 2nd coord z.
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| 53. Theorem. 2nd coord $U$ = $U$.
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| 54. Theorem.
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| If x and y are sets,
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| then 1st coord (x, y) = x
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| and 2nd coord (x, y) = y.
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| If either of x and y is not a set,
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| then 1st coord (x, y) = $U$
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| and 2nd coord (x, y) = $U$.
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| Proof. If x and y are sets, then the equality for
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| 1st coord is immediate from 50 and 51.
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| The equality for 2nd coord reduces to showing that
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| y = (x |^| y) |_| ((x |_| y) ~ x), by 50 and 52.
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| It is straightforward to see that
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| (x |_| y) ~ x = y ~ x,
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| and by the distributive law,
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| (y |^| x) |_| (y |^| ~x) = y |^| (x |_| ~x) = y |^| $U$ = y.
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| If either x or y is not a set, then, using 50 it is easy to compute
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| 1st coord (x, y) and 2nd coord (x, y). þ
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| 55. Theorem.
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| If x and y are sets and (x, y) = (u, v), then x = u and y = v.
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| JLK, Gen Top, page 259.
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| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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