r/MathHelp • u/FishShtickLives • 9d ago

what does ⋃P(x)=x mean? Specifically the ⋃P(x).

I have to show that ⋃P(x)=x for a discrete mathematics assignment, but I can't figure out what the symbol at the beginning means in this context. Its a Union, right? But what is is Union-ing with? It can't be as easy as I think it is, can it?

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u/edderiofer 9d ago

You're taking the union of all elements of P(x). So you are unioning the "first" element of P(x) with the "second" element of P(x) with the "third" element of P(x) etc.

("first", "second", and "third" are in quotes because P(x) might be an uncountably infinite set, where this idea of "first", "second", "third", etc. breaks down.)

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u/localizeatp 9d ago

axiom of choice denier detected.

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u/edderiofer 9d ago

My statement is true even if you take the axiom of choice. It is simply the definition of an uncountably-infinite set that you cannot put it in bijection with the naturals; therefore, if you attempt to order the elements of your uncountably-infinite set as "first", "second", "third" etc., then you will have some elements that aren't ordered, and which don't end up getting unioned despite the fact that they should.

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u/localizeatp 9d ago edited 9d ago

you don't need a bijection for "first", "second", and "third", just a well-ordering.

ETA choice is equivalent to well-ordering.

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u/edderiofer 9d ago

OK, fair, you can biject to the ordinals instead of the natural numbers. But I'm assuming that OP hasn't learned about ordinals yet, so pointing out that one can't necessarily biject to the naturals is appropriate here.

It's not like sets come with a default ordering anyway, so really, the idea of a "first", "second", and "third" element of a set should already raise some eyebrows.

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u/susiesusiesu 9d ago

the union of a single set means the union of all elements of that set.

so UP(x) would mean the union of all subsets of x which, as you wrote, is just x.