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Jan
17
comment Less suggestive terms for “vector addition” and “scalar multiplication”
It's also quite uncommon to denote a maximum with $\min$ ☺. I guess you wanted to write "minimum" here.
Jan
17
comment Let $2 = \{ 0, 1 \}$ and $X$, $Y$ be sets.
@GitGud: I disagree. The best definition of 2 is still "the successor of 1". Of course if you model the natural numbers with the von-Neumann construction, the successor of 1 turns out to be $\{0,1\}$. But that's already a result, not the definition.
Jan
17
comment Function with infinitely many right inverses?
Indeed, with that method you can easily show that the set of right inverses can have any cardinality, by just replacing $\mathbb N$ with a set of the desired cardinality.
Jan
17
comment a vectorspace, a linear map, the kernel and image of it
@Bernard: I don't think the Babylonians defined whether $0$ belongs to the set of natural numbers. Peano defined the natural numbers both without and with $0$, and used the notation $N_0$ for the latter. So there's precedent for that notation as old as the axiomatization of natural numbers themselves.
Jan
17
comment If the sets $A=\{x\in E, f(x)<\lambda\}$ and $B=\{x\in E, f(x)>\lambda\}$ are open, then $f$ is continuous
Strictly speaking, you haven't covered open sets like $(a,b)\cup (c,d)$. OTOH, the empty set is already covered by $(-\infty,b)\cap(a,\infty)$ through the case $a\ge b$.
Jan
17
comment Proving that basis always exists and is not unique
Of course to prove non-uniqueness for dimension $>1$ (the precondition to having two basis elements $e_1,e_2$ to begin with), you can also just replace $e_1$ with $e_1+e_2$. That works even for $\mathbb F_2$.
Jan
16
comment Is there a relationship between isometry as defined on metric spaces and those on vector spaces?
"I look up the definition of isometry online and many sources tell me it is a bijective "structure preserving" map." That sounds more like the definition of an isomorphism. Now an isometry is an isomorphism, but in general an isomorphism is not an isometry.
Jan
10
comment In Wikipedia's motivating example of wedge product, what happened to $e_1 \wedge e_1$
@Bye_World: So "alternating" does not mean the same as "antisymmetric"? I thought the property you seem to call "alternating" is called "nilpotent".
Jan
8
comment Counting how many possible images in an 800x500(3color) image.
Yes, as I wrote above: The most significant byte would be 1, and the 1,200,000 other bytes would be 0. However note that you only need 1,2000,000 bytes to represent $256^{800*500*3}$ different values (which would, obviously, not include the value $256^{800*500*3}$; the highest representable value in 1,200,000 bytes is $256^{800*500*3}-1$).
Jan
8
comment Counting how many possible images in an 800x500(3color) image.
Looking again at the three-digit example, interpreting the digits as number gives numbers from $0$ to $999$. That are $1000$ numbers, but the number $1000$ itself is not among them; indeed, it's the first one that's too large to fit in three digits. Another way to see it is: If it were only the numbers $1$ to $999$, you'd have $999$ numbers. But additionally you've got the $0$, and thus one more. But $999$ is already the largest 3-digit number, so you get a four-digit number. BTW, to write a positive integer $n$ in base $b$, you always need $\lfloor\log n/\log b\rfloor + 1$ digits.
Jan
8
comment Counting how many possible images in an 800x500(3color) image.
Well, that expression indeed gives exactly $1\,200\,000$. However, consider the number of sequences of 3 decimal digits. There are $10^3=1000$ such sequences, but the number $1000$ already has 4 digits (but $\ln 1000/\ln 10=3$).
Jan
8
comment Counting how many possible images in an 800x500(3color) image.
Actually, you need $1\,200\,001$ bytes to represent that number: The first byte is $1$, all others are $0$.
Jan
8
comment Is there a symbol for plus and minus as opposed to plus or minus?
@TitoPiezasIII: The closest I could come up with is Module[{R},R[s_]:=(2 b^3-9 a b c + s Sqrt[-4(b^2 - 3 a)])^(1/3); (2/3a)(R[+1]+R[-1])]
Jan
8
comment Prove that row rank of a matrix equals column rank
Why did you not add your answer to an existing question, instead of writing a duplicate question?
Jan
2
comment Is it possible to draw this picture without lifting the pen?
You omitted an important (but obvious) condition: The graph must be connected!
Dec
21
comment Have anyone ever thought of continuous analog Turing machine?
As far as I know, those analog computers were not universal computers, but purpose-built for specific problems. The point of the Turing machine is that it is universal.
Dec
20
comment $A \cap B$ = $A \cup B \iff A = B$
@dreamin: $A\subset A\cup B$ is a property of $\cup$. $A\cup B=A\cap B$ by assumption. $A\cap B\subset B$ is a property of $\cup$. Note that this proves only one direction, but the other direction works essentially the same.
Dec
19
comment A weighted measure of international diversity
What exactly is your definition of a "valid method"?
Dec
19
comment Description of the universe of sets
I think he is describing the von Neumann universe. In particular, I think with the "situation in which all the stages in the collection are completed" he refers to the stages corresponding to limit ordinals.
Dec
19
comment Problem solving rolling dice
What are the rules exactly? Does your partner tell you that you have rolled at least 9 as soon as this is true? Or could he remain silent until after you've done a few further rolls (after which you of course still have rolled at least 9)?