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 Oct 4 comment $1$ man can eat $1$ apple in $1$ day.How many apples can six men eat in six days? I can eat like twenty apples a day before I get sick. These puzzles are silently teaching you to view any relationship as (multi-)linear. Sure, work out the math that is intended for them, but also make sure to stay real, you know. If you can scramble five eggs in five minutes using one pan, how long does it take to scramble three eggs using four pans? Well, just use one of those pans to scramble them in five minutes as well! You know what I’m sayin’? Oct 3 comment Is $\emptyset$ in $R^n$ an open set? The symbol for the empty set is not Phi: $\phi ≠ ­­­­­­­\emptyset$ (\phi ≠ \emptyset). Oct 2 comment Sum of Odd Numbers make Squares Take any checkered sheet of paper and fill out the squares in the following way: Fill out any square, then fill out the adjacent L-shape of three squares, then fill out the adjacent L-shape of five squares … Ok, like indicated by Brian M. Scott. Oct 1 answered Intuition about open cover of $\mathbb Q$ of arbitrary small measure Oct 1 comment Intuition about open cover of $\mathbb Q$ of arbitrary small measure “They say …”, and it’s actually attributed to John von Neumann. Sep 25 answered Neighborhood base at the unit element in a topological group Sep 21 awarded Yearling Sep 19 comment Proof writing involving power set and cartesian product: $(P(A) \times P(B)) \subseteq P(A \times B)$ The statement would be true if by $(P(A) × P(B))$ you meant the set of all products $x × y$ with $x ∈ P(A)$ and $y ∈ P(B)$. Sep 17 comment If $E$ is the set of all infinite sequences of 0 and 1, $E$ equipotent to $E \times E$ “Let $(a,b) ∈ E × E$. Then $a = (a_1, a_2, a_3, …)$ for some $a_1, a_2, a_3, … ∈ \{0,1\}$ and $b = (b_1, b_2, b_3, …)$ for some $b_1, b_2, b_3, … ∈ \{0,1\}$. Let $x = (a_1, b_1, a_2, b_2, a_3, b_3, …)$. Then the image of $x$ under the given map (let’s say it’s called $φ$) is $$φ(x) = φ((a_1, b_1, a_2, b_2, a_3, b_3)) = ( (a_1, a_2, a_3, …), (b_1, b_2, b_3, …)) = (a,b).”$$ Sep 17 comment If $E$ is the set of all infinite sequences of 0 and 1, $E$ equipotent to $E \times E$ Zip up $({a_1}_j, {a_3}_j, {a_5}_j, \ldots)$ and $({a_2}_j, {a_4}_j, {a_6}_j, \ldots)$ to get …? Sep 17 comment If $E$ is the set of all infinite sequences of 0 and 1, $E$ equipotent to $E \times E$ You’re already done, to be quite frank. You can easily write down a preimage to the element you have written down. You already chose suggestive indexes. Sep 16 comment What is the usefulness of matrices? Matrices and matrix manipulations represent in a visually catchy way lots of stuff and corresponding stuff operations, most importantly linear maps and bilinear forms. Why I am saying this when it has been stated like a trillion times before me? (A) to help get the message across, (B) I want to stress that matrix repersentation is visually catchy and the fact that they also nicely represent operations on stuff (like map composition by matrix multiplication). Sep 12 awarded Popular Question Sep 3 awarded Good Answer Aug 24 comment May Algebraic Geometry be appropriate for me? @PeteL.Clark The differences in role between letters and concepts I mentioned were supposed to show that hating certain letters is fundamentally und utterly different to hating certain concepts. Aug 23 comment May Algebraic Geometry be appropriate for me? By the way, Majin-Bu, out of curiosity: What happens if you think of the polynomial ring $R[X]$ of a commutative ring $R$ as the free commutative $R$-algebra over $X$? Like viewing polynomials as the left adjoint $R[•]\colon \mathrm{Set} → R\mathrm{-CAlg}$ to the forgetful functor? Aug 23 comment May Algebraic Geometry be appropriate for me? … 3) letters all share the same functionality, 4) concepts do come with a way to think, and 5) disliking concepts (like zero, imaginary numbers, set theory, general nonsense) has been done by professional mathematicians. Therefore, to me it doesn’t seem to be that much immature to hate a certain mathematical concept as a mathematician. Is it yet? Aug 23 comment May Algebraic Geometry be appropriate for me? @PeteL.Clark Saying “sounds like a would-be writer averring that he we try to avoud isng the letter “g” as much as possible” – isn’t that a bit overstating it a bit? This is meant as a real question, because I’m not a professional mathematician, but I don’t see the analogy “letters – concepts” at all: 1) Words are used by writers as material, concepts are used by mathematicians as tools, and letters are “submaterial”, writers actually don’t use letters. 2) you cannot introduce new letters to better express yourself as a writer, letters are sort of determined once and for all, and … Aug 23 comment Existence of holomorphic function from unit disc to itself. I’m not sure, it might help to check out Schwarz Lemma and the characterization of automorphisms of the unit disc (which are certain Möbius transformations). Aug 22 comment Existence of holomorphic function from unit disc to itself. Well, yes (if $a ≠ 0$), since then for $\lvert z \rvert < 1$, you’d have $\lvert f(z) \rvert ≤ \lvert az \rvert + \lvert b\rvert < \lvert a \rvert + \lvert b \rvert ≤ 1$. But this doesn’t help you here, because such an $f$ has $f’ = a$, so condition “$f’(3/4) = 1$” forces $a = 1$, and $\lvert a \rvert + \lvert b \rvert ≤ 1$ then forces $b = 0$, so $f = \mathrm{id}$, but that doesn’t do the trick with ”$f(1/2) = -1/2$”.