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What is {ℝ}?

  1. Feb 6, 2013 #1
    What is {ℝ}???

    Someone I know tried to convey me the meaning of {[itex]ℝ[/itex]}, stating it represents a set of real numbers. But using notation, {[itex]ℝ[/itex]}, is implying that the real space is (improperly) contained in a set, and I don't think this makes any logical sense.
    On the other hand, we can say {[itex]x \in S | \forall S \in ℝ[/itex]}, etc....or simply [itex]x \in ℝ[/itex].
    Another way of thinking about this is instead of putting your foot in a sock, you are putting the sock into your foot and it's disturbing.

    Am I right or wrong?

  2. jcsd
  3. Feb 6, 2013 #2


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    Re: What is {ℝ}???

    It is perfectly valid to put sets inside of other sets. For example, we may consider the set of all subsets of the complex numbers ##\mathbb{C}##. This is called the power set of ##\mathbb{C}## and is sometimes given the notation ##\mathcal{P}(\mathbb{C})## or ##2^\mathbb{C}##. Any subset of ##\mathbb{C}## is an element of ##\mathcal{P}(\mathbb{C})##. For example, we have ##\mathbb{R} \subset \mathbb{C}## and ##\mathbb{R} \in \mathcal{P}(\mathbb{C})##. We can form subsets of ##\mathcal{P}(\mathbb{C})## in the usual way, by putting elements of ##\mathcal{P}(\mathbb{C})## into a set. Thus ##\{\mathbb{Z}, \mathbb{Q}, \mathbb{R}\} \subset \mathcal{P}(\mathbb{C})##, and a special case is a subset containing only one set, such as your example: ##\{\mathbb{R}\} \subset \mathcal{P}(\mathbb{C})##.
  4. Feb 7, 2013 #3
    Re: What is {ℝ}???

    Thanks bjunniii, you have a good point. However, let me rephrase my doubt.

    We want to use a notation to represent a set of all real number, say [itex]X[/itex]. It is immediately apparent that [itex]x \in X \subseteq ℝ[/itex] for some real number [itex]x[/itex]. In this case, we are not not considering any stronger set, for instance, [itex]P(ℂ)[/itex] as you mentioned.
    Now having limited ourselves to real space, it is rather redundant to say the set is represented as {[itex]ℝ[/itex]} because since [itex]ℝ[/itex] is not a proper subspace in this case. This is the reason why I said "this does not make any logical sense"; I am ridiculed by those curly brackets!

    Instead, we could simply write [itex]X \in ℝ[/itex] that give a much more direct and sensible idea of what space we are talking about.

    Do you agree?
  5. Feb 7, 2013 #4
    Re: What is {ℝ}???

    The set [itex]\{\mathbb{R}\}[/itex] is a set which contains only one element. Its element is [itex]\mathbb{R}[/itex]. There is no reason why such a construction would not be allowed.
    It is true, however, that sets like [itex]\{\mathbb{R}\}[/itex] don't play a big role in mathematics.
  6. Feb 7, 2013 #5


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    Re: What is {ℝ}???

    Unless you state otherwise, all set theory is done in ZFC, and one of its axioms is the axiom of pairing. It asserts: given any two sets A and B, there exists set C with exactly those two elements, i.e. C = {A, B}.

    So it does logically exist (in this case A = B = ℝ).
  7. Feb 7, 2013 #6
    Re: What is {ℝ}???

    @micromass, @pwsnafu, I see what you are saying. Overall, you have convinced me {[itex]ℝ[/itex]} is possible.

    But, The notation with curly bracket is directly defining the single element in this set as the real space which is essentially another set. I was simply saying there is no necessity to put real space as a subset of a set in the first place; there are no other disjoint elements.
  8. Feb 8, 2013 #7


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    Re: What is {ℝ}???

    I was about to write "what's your point?" until I re-read the original post:

    Your friend is wrong. The set of real numbers is ℝ. (I bold "the" because there is only one.)

    But ##\{\mathbb{R}\} \neq \mathbb{R}##. The left hand side is "a set with one element, and that element is ℝ". They are not the same thing.
  9. Feb 9, 2013 #8
    Re: What is {ℝ}???

    Aha, there we go. Thanks alot pwsnafu for the verification! as well as others for your time generous reply.
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