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Jul
13
revised What does “extend linearly” mean in linear algebra?
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Jul
13
comment What does “extend linearly” mean in linear algebra?
@Jack: sorry! It is a common shortcut for "linearly independent". I edited it out.
Jul
13
revised What does “extend linearly” mean in linear algebra?
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Jul
13
answered What does “extend linearly” mean in linear algebra?
Jul
12
revised What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?
added 404 characters in body
Jul
12
answered What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?
Jul
12
asked Models of hyperbolic geometry
Jul
12
comment Problem from Halmos's Finite Dimensional Vector Spaces
A note to your note: well, functionals are functions, in the usual set-theoretic meaning! So the notation $y(x)$ for a functional $y\in P'$ and a vector $x\in P$ is the same notation as ever.
Jul
11
comment What does {2x|P(x)} mean?
meaning a priori, that is, unless the $x$ you are talking about are elements of a structure where "multiply by 2" makes sense: for example, the natural numbers $\mathbb{N}$, the real numbers $\mathbb{R}$, the continuous functions $\mathbb{R}\to \mathbb{R}$, etc.
Jul
11
comment What does {2x|P(x)} mean?
I don't understand your objection (the latex in it was ill-rendered). Are you objecting to the use of $x\in \dots$? Think that a priori $2x$ doesn't have a meaning. For example, if $x$ is a color, such as red, what does $2x$ mean? It does have a meaning if $x$ is an element of some mathematical structure that admits to "multiply an element by 2", for example a monoid, a group, a ring, a field, a vector space... Thus, in general set theory, a set $\{2x: P(x)\}$ (note that I am using $:$ instead of $|$: it is different notation for the same thing, see Bill's answer below) does not have..
Jul
11
comment What does {2x|P(x)} mean?
@fahad: No, the order in which you check the propositions is not relevant: $\{y:\exists x (y=2x \wedge P(x))\}=\{y: \exists x (P(x) \wedge y=2x\}$. In other words, conjunction is commutative.
Jul
11
comment What does {2x|P(x)} mean?
Just as a comment, you should be aware that a "collection of elements" described in such manner is not necessarily a set-- see e.g. en.wikipedia.org/wiki/Russell%27s_paradox . However, if in this case you are thinking of $x\in \mathbb{R}$, then this set exists, but to be sure it exists you should write it as $\{y\in \mathbb{R}: \exists x (x\in \mathbb{R} \wedge y=2x \wedge P(x))\}$ and use the fact that $\mathbb{R}$ exists and the comprehension axiom-- see: en.wikipedia.org/wiki/Comprehension_axiom
Jul
9
comment Why isn't the perfect closure separable?
Thank you for some great insight !
Jul
8
revised Problem in understanding set theory
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Jul
8
comment Problem in understanding set theory
Well, in your answer to 2), you're not really proving that it is never true that $A\subseteq B$, you're just giving an example where it is not true...
Jul
8
comment Problem in understanding set theory
It would be necessary for you to specify under which set theory you are working. In my answer below I assumed you were dealing with Zermelo-Fraenkel (ZF).
Jul
8
revised Problem in understanding set theory
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Jul
8
comment Problem in understanding set theory
@Gerry: nicely put. I was going to comment it in this form: "A bag is not the same as a bag containing a bag".
Jul
8
revised Problem in understanding set theory
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Jul
8
answered Problem in understanding set theory