NuraShik
(X∪Y)=X+Y−(X∩Y) holds true for any set X and Y? Here, X + Y means all elements in X and Y including repeated elements. elementary-set-theory Share Cite Follow asked Apr 12, 2015 at 22:30 NatureDevil's user avatar NatureDevil 12111 bronze badge Better to say that n(X∪Y)=n(X)+n(Y)−n(X∩Y). – tomi Apr 12, 2015 at 22:35 2 You need to be precise about how you model sets with repetition, and then how you define the set operations, and the meaning of equality. Before you do that, the question is too vague (though certainly there is a grain of truth to it). – Ittay Weiss Apr 12, 2015 at 22:38 Add a comment 1 Answer Sorted by:
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Not quite literally true, because in set theory there is no such thing as "repeated elements". So + becomes just ∪, in which case there is no unique reading to the statement (do you take ∪ before − or after?) and that makes the statement a bit vague.
However you can change it slightly and then it is true: X∪Y=X∪(Y−(X∩Y))
If you want to talk about finite sets, then cardinal arithmetic makes this statement true, namely: |X∪Y|=|X|+|Y|−|X∩Y| But the finiteness assumption is somewhat necessary here, since subtraction is not well-defined for infinite sets. Share Cite Follow answered Apr 12, 2015 at 22:35 Asaf Karagila's user avatar Asaf Karagila♦ 388k4444 gold badges592592 silver badges10001000 bronze badges But |X|+|Y|=|X∪Y|+|X∩Y| for arbitrary sets X,Y holds in ZF, right? And the same goes for the general in-and-out formula, written without minus signs? – bof Apr 12, 2015 at 22:54 1 Yes, that is correct. – Asaf Karagila ♦ Apr 12, 2015 at 22:56
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artem076741
Мне нужно пошаговое решение, а не копипаст непонятно откуда
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elementary-set-theory
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asked Apr 12, 2015 at 22:30
NatureDevil's user avatar
NatureDevil
12111 bronze badge
Better to say that n(X∪Y)=n(X)+n(Y)−n(X∩Y). –
tomi
Apr 12, 2015 at 22:35
2
You need to be precise about how you model sets with repetition, and then how you define the set operations, and the meaning of equality. Before you do that, the question is too vague (though certainly there is a grain of truth to it). –
Ittay Weiss
Apr 12, 2015 at 22:38
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Sorted by:
1
Not quite literally true, because in set theory there is no such thing as "repeated elements". So + becomes just ∪, in which case there is no unique reading to the statement (do you take ∪ before − or after?) and that makes the statement a bit vague.
However you can change it slightly and then it is true:
X∪Y=X∪(Y−(X∩Y))
If you want to talk about finite sets, then cardinal arithmetic makes this statement true, namely:
|X∪Y|=|X|+|Y|−|X∩Y|
But the finiteness assumption is somewhat necessary here, since subtraction is not well-defined for infinite sets.
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answered Apr 12, 2015 at 22:35
Asaf Karagila's user avatar
Asaf Karagila♦
388k4444 gold badges592592 silver badges10001000 bronze badges
But |X|+|Y|=|X∪Y|+|X∩Y| for arbitrary sets X,Y holds in ZF, right? And the same goes for the general in-and-out formula, written without minus signs? –
bof
Apr 12, 2015 at 22:54
1
Yes, that is correct. –
Asaf Karagila
♦
Apr 12, 2015 at 22:56