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forum.math.uoa.gr • Προβολή θέματος - Ισόμορφος με δυναμοσύνολο

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Ημερομηνία 23 Νοέμ 2017, 22:38

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 Θέμα δημοσίευσης: Ισόμορφος με δυναμοσύνολο
ΔημοσίευσηΔημοσιεύτηκε: 12 Ιούλ 2016, 13:27 
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Εγγραφή: 10 Μαρ 2016, 14:08
Δημοσ.: 33
Γεια σας. Μια πολύ όμορφη αλλά και εξαιρετικά δύσκολη άσκηση. Μου άρεσε και την μοιράζομαι.

Έστω \displaystyle{R} ένας πεπερασμένος δακτύλιος του [unparseable or potentially dangerous latex formula] . Αποδείξτε ότι

υπάρχει μη κενό σύνολο [unparseable or potentially dangerous latex formula] τέτοιο, ώστε [unparseable or potentially dangerous latex formula] ως δακτύλιοι.

Σχόλιο: Το δυναμοσύνολο [unparseable or potentially dangerous latex formula] ενός μη κενού συνόλου [unparseable or potentially dangerous latex formula]

αποκτά δομή δακτυλίου με πρόσθεση την συμμετρική διαφορά συνόλων και πολλαπλασιασμό την τομή συνόλων.

Το μηδενικό στοιχείο είναι το κενό σύνολο ενώ η μονάδα είναι το σύνολο [unparseable or potentially dangerous latex formula] .


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 Θέμα δημοσίευσης: Re: Ισόμορφος με δυναμοσύνολο
ΔημοσίευσηΔημοσιεύτηκε: 12 Ιούλ 2016, 23:30 
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Εγγραφή: 11 Δεκ 2014, 19:45
Δημοσ.: 49
Γειά σου Βαγγέλη,

υποπτεύομαι (χωρίς να το έχω αποδείξει) ότι αν θέσεις X το σύνολο των ατόμων της άλγεβρας Boole που επάγεται από τη R, τότε [unparseable or potentially dangerous latex formula]. Αυτό είχες στο μυαλό σου; Αν όχι, τότε μην μας αποκαλύψεις ακόμα τη λύση!

_________________
"Problems worthy of attack prove their worth by fighting back" -Paul Erdős


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 Θέμα δημοσίευσης: Re: Ισόμορφος με δυναμοσύνολο
ΔημοσίευσηΔημοσιεύτηκε: 13 Ιούλ 2016, 11:21 
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Εγγραφή: 10 Μαρ 2016, 14:08
Δημοσ.: 33
Καλημέρα Βασίλη. Δεν είχα αυτό στο μυαλό μου.

Μπορείς να ορίσεις το σύνολο των ατόμων της άλγεβρας Boole που επάγεται από τον \displaystyle{R} ;

Δεν το έχω ξαναδει αυτό.


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 Θέμα δημοσίευσης: Ισόμορφος με δυναμοσύνολο
ΔημοσίευσηΔημοσιεύτηκε: 23 Ιούλ 2016, 21:40 
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Regular Forumer

Εγγραφή: 11 Φεβ 2007, 21:13
Δημοσ.: 630
Πρέπει να δείξουμε λοιπόν : [unparseable or potentially dangerous latex formula]
Για το σύνολο [unparseable or potentially dangerous latex formula] ορίζουμε την χαρακτηριστική συνάρτηση τού [unparseable or potentially dangerous latex formula], δηλαδή

[unparseable or potentially dangerous latex formula]

Τότε η απεικόνιση [unparseable or potentially dangerous latex formula] που σε κάθε στοιχείο [unparseable or potentially dangerous latex formula] από το [unparseable or potentially dangerous latex formula] αντιστοιχίζει την χαρακτηριστική του συνάρτηση [unparseable or potentially dangerous latex formula], είναι ένας ισομορφισμός.
Η ερριπτικότητα της \Phi είναι προφανής
Εάν τώρα [unparseable or potentially dangerous latex formula] και [unparseable or potentially dangerous latex formula], τότε ισχύει [unparseable or potentially dangerous latex formula] που καταδεικνύει την επιρριπτικότητα. Τελικά ισχύει

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

δηλαδή επίσης
[unparseable or potentially dangerous latex formula] και [unparseable or potentially dangerous latex formula]


ΥΣ
Πρέπει να διευκρινίσω ορισμένα σημεία για να φανεί η ιδέα της απόδειξης...
1)
[unparseable or potentially dangerous latex formula], είναι το σύνολο όλων των απεικονίσεων [unparseable or potentially dangerous latex formula], απο το τυχαίο σύνολο [unparseable or potentially dangerous latex formula] στο [unparseable or potentially dangerous latex formula]
Γιά κάθε ζευγάρι απεικονίσεων ισχύει τὀτε:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Τότε είναι το
[unparseable or potentially dangerous latex formula] ένας δακτύλιος του Boole, o B(X)

2)
Γιά τα σύνολα [unparseable or potentially dangerous latex formula] έχουμε :
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
οπότε το [unparseable or potentially dangerous latex formula] γίνεται δακτύλιος τού Boole, ο [unparseable or potentially dangerous latex formula], όπου το κενό σύνολο \varnothing
και το σύνολο [unparseable or potentially dangerous latex formula] το ίδιο παίζουν τον ρόλο τών ουδετέρων στοιχείων Μηδέν και Ένα.


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