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Jul
14
answered Can the indexing set of a cartesian product have the cardinality of the continuum?
Jul
14
comment How to add compound fractions?
I'd like to point out, that, however, $x= \frac{1}{\frac{1}{x}}$ only when $x\not=0$.
Jul
14
comment Mixing Problem: gallons in tank at a time t
@Zev: good edit, I had read "galois in tank at a time $t$" which was slightly disturbing! :D
Jul
14
awarded  Civic Duty
Jul
13
revised What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?
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Jul
13
comment What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?
@Asaf: have you read my answer thoroughly? I'm not saying that $A^I$ makes no sense for infinite $I$ -- on the contrary! If you are referring to the last line, I'm saying that defining $A^I$ as a set of tuples, for infinite $I$, makes no sense a priori. But of course, as a set of functions, sure.
Jul
13
comment What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?
@Asaf: they can't be defined on a first year undergraduate course :)
Jul
13
comment What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?
@confused: I've added a couple of things which may be nice to you. To think of $\mathbb{N}^2$ as pairs of natural numbers is the same that thinking $\mathbb{N}^2$ as the set of functions $2\to \mathbb{N}$, since $2=\{0,1\}$ in ordinary set theory. This is easily generalized.
Jul
13
revised What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?
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Jul
13
comment What does “extend linearly” mean in linear algebra?
@Jack: of course not! My bad.
Jul
13
revised What does “extend linearly” mean in linear algebra?
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Jul
13
revised What does “extend linearly” mean in linear algebra?
added 100 characters in body
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?
added 352 characters in body
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.