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December 5

Analysis Q

Define $f(n):=\sum _{k=1}^{n}\ln(n/k)$. Show that $f(n)=O(n)$.

--AnalysisAlgebra (talk) 01:03, 5 December 2012 (UTC)

See Stirling's approximation. Sławomir Biały (talk) 01:09, 5 December 2012 (UTC)

Natural solutions of the equation: $a_{1}^{a_{2}^{a_{3}}}=b_{1}^{b_{2}^{b_{3}}}$, for $\{a_{1},a_{2},a_{3}\}=\{b_{1},b_{2},b_{3}\}$, while $a_{1}\neq b_{1},a_{2}\neq b_{2},a_{3}\neq b_{3}$.

Any suggestions? HOOTmag (talk) 10:01, 5 December 2012 (UTC)

Loads like {2,2,3}={4,4,1} Dmcq (talk) 10:50, 5 December 2012 (UTC)

Are you sure the two sets are equal...? --CiaPan (talk) 10:58, 5 December 2012 (UTC)

1^{n^0} = n^{0^1}.--80.109.106.49 (talk) 11:33, 5 December 2012 (UTC)

No, I've defined explicitely: "natural" solutions, rather than "whole" solutions. So if this was not clear, I'll make it clear - right now: I'm referring to positive integers only. HOOTmag (talk) 11:43, 5 December 2012 (UTC)

There is not universal agreement on the definition of "natural number". Often "natural" and "whole" are used in the reverse of the way you use it. If it's important, you should specify (e.g. "positive integer").--80.109.106.49 (talk) 15:32, 5 December 2012 (UTC)

I have already stated: "If this was not clear, I'll make it clear - right now: I'm referring to positive integers only". HOOTmag (talk) 19:47, 5 December 2012 (UTC)

There's a few infinite sequences of solutions:

• 0**(a**b) = 0**(b**a) = 0
• 1**(a**b) = 1**(b**a) = 1
• a**(0**b) = b**(0**a) = 1
• 1**(a**0) = a**(0**1) = 1
• 1**(a**b) = 1**(b**a) = 1

There's also this lone solution:

• 2**(4**1) = 4**(2**1)

And the symmetric solution for each where you swap a with b.

I don't know whether there are any other solutions, but there don't seem to be any small solutions at least.

– b_jonas 12:15, 5 December 2012 (UTC)

Update: ok wait, you have extra restrictions I've ignored that exclude all the solutions above. In that case, I don't know whether there are any solutions at all. – b_jonas 12:18, 5 December 2012 (UTC)

Suggestion: KISS.
You might equivalently say 'find natural a, b and c, pairwise different, such that $a^{b^{c}}=b^{c^{a}}$'.
CiaPan (talk) 12:51, 5 December 2012 (UTC)

But I don't think CiaPan's restatement captures all the possibilities allowed by the OP's problem statement: What about $c^{a^{b}}$, $c^{b^{a}}$, $b^{a^{c}}$, and $a^{c^{b}}$? It seems to me there are 6 distinct forms, and the OP's question is whether any two of them (with pairwise different a, b, c) are equal. So I think the OP's problem specification is the simplest one, and it has the advantage of pointing out clearly the symmetry of the problem. Duoduoduo (talk) 15:03, 5 December 2012 (UTC)

CiaPan's restatement is equivalent. $c^{b^{a}}$ is disallowed, because the OP requires $a_{2}\neq b_{2}$. Similarly $b^{a^{c}}$ and $a^{c^{b}}$. If $a^{b^{c}}=c^{a^{b}}$, then $a'^{b'^{c'}}=b'^{c'^{a'}}$, under the map $a'=c$, $b'=a$, $c'=b$.--80.109.106.49 (talk) 15:39, 5 December 2012 (UTC)

Why are $c^{b^{a}}$ and $b^{a^{c}}$ disallowed? What about $c^{b^{a}}=b^{a^{c}}$? Or is that too a transformation of CiaPan's formulations? Remember that his formulation needs to allow for equality between any two of the 6 ways of writing it as long as those two obey the inequality constraints. Duoduoduo (talk) 17:29, 5 December 2012 (UTC)

In the original formulation, $a^{b^{c}}=c^{b^{a}}$ would make $a_{2}=b=b_{2}$, which is not allowed, and similarly for $a^{b^{c}}=b^{a^{c}}$. As for $c^{b^{a}}=b^{a^{c}}$, yes, this is a transformation of CiaPan's formulation. Just rename the variables.—Emil J. 18:11, 5 December 2012 (UTC)

The original statement does not imply that a, b, c are pairwise distinct. What about $a^{b^{b}}=b^{a^{a}}$, $a^{b^{a}}=b^{a^{b}}$, $a^{a^{b}}=b^{b^{a}}$.—Emil J. 15:47, 5 December 2012 (UTC)

No, the sets {a, a, b} and {a, b, b} are not equivalent, as required by the question. In fact the question does require three distinct numbers. Duoduoduo (talk) 17:12, 5 December 2012 (UTC) Proof: if two members of the set are the same, say k, then how are you going to fill in the three slots in two different ways such that no slot has k as its entry for both expressions? Duoduoduo (talk) 17:21, 5 December 2012 (UTC)

You and I must be using a different definition of a set: {a, a, b} = {a, b} = {a, b, b}. Anyway, now the OP has given yet another formulation, which is equivalent neither to the original statement nor to CiaPan’s reformulation, as it does not require any of the numbers to be distinct.—Emil J. 17:56, 5 December 2012 (UTC)

(ec)The title of the post says $\{a_{1},a_{2},a_{3}\}=\{b_{1},b_{2},b_{3}\}$. It seems pretty clear to me that the intent of that was "counting multiplicities". In other words, the sets are element by element identical up to ordering, with 3 not necessarily distinct elements. Duoduoduo (talk) 18:11, 5 December 2012 (UTC) That is, not necessarily distinct until the inequality constraints are considered. Duoduoduo (talk) 18:15, 5 December 2012 (UTC)

The sets {a, a, b} and {a, b, b} are identical, so my formulation is not equivalent to CiaPan's, which only covers the case of three distinct numbers, while not covering the other three cases indicated by Emil, i.e. $x^{x^{y}}=y^{y^{x}}$, and $x^{y^{y}}=y^{x^{x}}$, and $x^{y^{x}}=y^{x^{y}}$. HOOTmag (talk) 13:19, 6 December 2012 (UTC)

Wrong, the original statement does imply that a, b, c are pairwise distinct: it explicitly requires $a_{i}\neq b_{i}$ for each $i\in \{1,2,3\}$ which could not be satisfied by any permutation $(b_{i})$ of three-terms sequence $(a_{i})$ if any two a terms were equal. --CiaPan (talk) 09:01, 6 December 2012 (UTC)

You didn't answer Emil's last question: What about $x^{x^{y}}=y^{y^{x}}$, and $x^{y^{y}}=y^{x^{x}}$, and $x^{y^{x}}=y^{x^{y}}$? Note that all of these cases do fulfil the requirement: $a_{i}\neq b_{i}$ for each $i\in \{1,2,3\}$. HOOTmag (talk) 13:19, 6 December 2012 (UTC)

Ok, so find three different positive integers x,y,z, satisfying: $x^{y^{z}}=y^{z^{x}}$, or two different positive integers x,y, satisfying: $x^{y^{y}}=y^{x^{x}}$, or $x^{x^{y}}=y^{y^{x}}$, or $x^{y^{x}}=y^{x^{y}}$. HOOTmag (talk) 16:18, 5 December 2012 (UTC)

EmilJ says above now the OP has given yet another formulation, which is equivalent neither to the original statement nor to CiaPan’s reformulation, as it does not require any of the numbers to be distinct. HOOTmag, did you mean to include a condition that x≠y, y≠z, and z≠x ? Duoduoduo (talk) 18:19, 5 December 2012 (UTC)

Note that the condition that x≠y, y≠z, and z≠x, only refers to the case of three different positive integers x,y,z, satisfying $x^{y^{z}}=y^{z^{x}}$. Yes, I forgot to add this condition, so I've just added it now, and I've just added also the other three cases of two different x,y. HOOTmag (talk) 19:14, 5 December 2012 (UTC)

It's easy to find solutions to $2^{2^{n}}=4^{4^{m}}$, and this should point the way to other large classes of solutions. Looie496 (talk) 16:53, 5 December 2012 (UTC)

No, the question requires the two sets to be identical. {2, 2, n} and {4, 4, m} are not identical for any m, n. Duoduoduo (talk) 17:12, 5 December 2012 (UTC)

There are no solutions to $a^{b^{c}}=b^{c^{a}}.$ We know from a recent thread that the only solution to $x^{y}=y^{x}$ is $2^{4}=4^{2}.$ So collate the left sides and the right sides: the left side gives a=2, $b^{c}$=4, and the right side gives b=4, $c^{a}=2$. So the right side implies c=2, a=1, and with a=1 the left side equals 1 whereas the right side is 4 raised to a power. Duoduoduo (talk) 17:43, 5 December 2012 (UTC) Striking my own nonsense: this problem does not require $x^{y}=y^{x}$. Duoduoduo (talk) 17:50, 5 December 2012 (UTC)

OK, let's add a 4th variable to the original question. While the technique above and induction would certainly rule out any examples of a^b^c^d = b^c^d^a, is there anything more general so that it can be proved that a^b^c^d = b^d^a^c with all different?Naraht (talk) 17:48, 5 December 2012 (UTC)

Isn't here anybody who can answer my trivial question, which can be undertsood even by teens? HOOTmag (talk) 12:52, 6 December 2012 (UTC)

That's the problem -- most or all of us here who are trying to help you are not teens! Your characterization can be undertsood (sic) even by teens is obviously wrong, judging by the above discussion in which different people have interpreted things in different ways, and into which you've injected another formulation that turned out to lack an intended set of restrictions. You need to get things clear the first time, and get them clear again when you restate, and then not insult people who are trying to help for not understanding something that supposedly even a teen could understand. Duoduoduo (talk) 16:46, 6 December 2012 (UTC)

I can't figure out why you have to think something here may "insult". Yes, I know you are not teens. I just said that my question can be understood even by teens, and of course by all of you (who are not teens), and all of you did understand my question, as I stated it the first time. What wasn't clear in my first formulation? What do you mean by "You need to get things clear the first time"? Btw, my second formulation is clear as well, and it's equivalent to the first one (although it wasn't equivalent in the beginning, but I have already fixed it). HOOTmag (talk) 23:51, 6 December 2012 (UTC)

To rephrase, the problem appears to seek solutions to any of the following four equations, where distinct variables are constrained to be distinct positive integers:

a^bc = b^ca, a^ba = b^ab, a^ab = b^ba, a^bb = b^aa.

The fourth equation can be eliminated (it is easy to show that for a real solution with a < b, 1 < a < 1.5, and a is thus not an integer). The second and third cases may be susceptable to a similar approach, just using graphs on two variables. In the first case it may pay to consider factors. For example, it is straightforward to show that a = x^p and b = x^q for some integers x > 1, p, q. It seems rather likely that the answer is that there are no solutions. — Quondum 05:06, 8 December 2012 (UTC)

I only agree about the last three cases:

• As for the last three cases, I agree, but your reasoning is insufficient (it ignores some important cases). The following reason is sufficient: a and b are known to be distinct, so one of them, say b, is larger than a, so it's easily provable that: b^ab > a^ba, b^ba > a^ab, a^bb > b^aa, unless a=1 in which case b^aa > a^bb.
• However, I don't agree about the first case a^bc = b^ca, unless c>b>a or c=1, in which case it's easily provable that a^bc > b^ca; but if a>b>1, then without knowing the value of c you cannot determine in advance whether a^bc < b^ca (e.g. as 4²³ < 2³⁴ ) or a^bc > b^ca (e.g. as 4²¹⁵ > 2¹⁵⁴ ), so I don't see how you can determine in advance whether a^bc = b^ca.
• Btw, I don't agree with what you've claimed that: a = x^p and b = x^q for some integers x > 1, p, q. How about a and b that don't have a common factor? HOOTmag (talk) 18:38, 8 December 2012 (UTC)

Nice to have all but one equation eliminated. In answer to your last question, the prime factors of the expression on the left are exactly those of a, and similarly on the right of b in a^bc = b^ca. Therefore a and b have exactly the same prime factors, but not necessarily the same multiplicity of each. However, you will see that the multiplicities of the prime factors must be in the same ratio in a and b (they are each put to an integer power, which multiplies the number of each prime factor), hence the conclusion a = x^p and b = x^q for some positive integers x, p, q. x provides all the prime factors in the correct ratio of multiplicities. x ≠ 1, p ≠ 0, q ≠ 0 because otherwise a = b, which would violate the premise. — Quondum 21:16, 8 December 2012 (UTC)

All right, but how does all of that help you determine in advance whether a^bc = b^ca? Note that if a>b>1, then without knowing the value of c you cannot even determine in advance whether a^bc < b^ca (e.g. as 4²³ < 2³⁴ ) or a^bc > b^ca (e.g. as 4²¹⁵ > 2¹⁵⁴ ), so I don't see how your claim about the common factors of a,b can help you determine in advance whether a^bc = b^ca. HOOTmag (talk) 22:57, 8 December 2012 (UTC)

That one's easy. Write N=b^c, and M = c^a. We know that N and M are both positive, and a^N = b^M. It follows immediately that, writing K = GCD(N,M), that there is an x such that a = x^M/K, and b = x^N/K. — Arthur Rubin (talk) 00:33, 9 December 2012 (UTC)

I agree it doesn't help much, but a = x^p, b = x^q, and p b^c = q c^a (or p x^{q c} = q c^xp) might provide some reduction, with p and q relatively prime. — Arthur Rubin (talk) 00:36, 9 December 2012 (UTC)

How do all of these facts you've indicated above help you determine whether there are distinct positive integers a,b,c satisfying: a^bc = b^ca? HOOTmag (talk) 01:40, 9 December 2012 (UTC)

Look at Aurthur's last (parenthesized) equation above. The left clearly has several factors of x, regardless of whether p has such factors, since q ≥ 1, c > 1. This means that the right has several, either in q or in c or both. q cannot account for all these factors. (Pretty obviously: if q = 1, it doesn't. If q > 1, then it forces (exponentially) more factors of x than it can account for.) Therefore, c also has x as a factor. Essentially, this all constitutes exploration, and it seems there are various rather tight constraints to be found. However, just as with Fermat's Last Theorem, solutions of even an innocuous Diophantine equation (and this is an exponential Diophantine equation) can be tricky to resolve. Is it more than an "I wonder" sort of proposition? — Quondum 06:58, 9 December 2012 (UTC)

To sum up, what you've only proved is that my Diophantine equation constitutes exploration (btw I knew that from the very beginning), but I still wonder (as I've always wondered from the very beginning) whether this "tricky" Diophantine equation really has a solution. Note that also the Diophantine equation x^y=y^x (for two distinct integers x,y) constitutes exploration, and this equation really has a solution (2,4).

Btw, since for every real a>b≥2 the function f(x)=a^bx-b^xa must cut the X axis at a unique point, therefore my question entails the following question: can be restated (in an equivalent form) as whether, for any two given distinct positive integers a>b≥2, that point (at which the function f(x)=a^bx-b^xa cuts the X axis), is a third positive integer. HOOTmag (talk) 08:53, 9 December 2012 (UTC)

"[S]ince for every real a>b≥2, f(x)=a^bx-b^xa must cut the [positive] X axis at a unique point...." . Also, it's not clear that a>b is relevant to the original problem. Thinking about it, you're probably wrong....

1. $a^{b^{x}}:b^{x^{a}}$
2. $b^{x}\ln a:x^{a}\ln b$
3. $x\ln b+\ln \ln a:a\ln x+\ln \ln b$
   • At x = 0, $\ln \ln a>-\infty$
   • At x = $\infty$, lhs > rhs
   • The derivative of the difference is 0 only at $x={\frac {a}{\ln b}}$.
   • Hence there are 0 or 2 positive solutions (or exactly one if $x={\frac {a}{\ln b}}$ is a solution).
   • At that point, we have $a+\ln \ln a:a\ln a-(a-1)\ln \ln b$, which goes to the rhs if a>b>1 and for some larger values of b. For those values of a and b, there are then 2 positive solutions. — Arthur Rubin (talk) 07:25, 10 December 2012 (UTC)

.

"[I]t's not clear that a>b is relevant to the original problem". No, It's really relevant (indirectly though), because one can very easily prove that if c>b>a≥2 then the function a^bx-b^xa must be positive, hence cannot cut the X axis.

.

"Thinking about it, you're probably wrong....". Wrong about what? Having zero solutions (not smaller than 2) is only relevant in a case I was not talking about, i.e. when c>b>a≥2 (see above why); Whereas a second positive solution is only possible in the range (0,2), which is irrelevant to the three numbers we're talking about, which can't be smaller than 2 (see above why, in the second section of my response to Quondum's first response).

.

Just try to think about that, in some simple examples: e.g. if a=3 and b=2, then the only real solution (not smaller than 2) must be somewhere in the range (4,10), because 3²⁴-2⁴³<0, but 3²¹⁰-2¹⁰³>0, whereas the function 3^2x-2^x3 is continuous and monotonic in the relevant range (i.e. from 2 to infinity). Where is the second real solution (not smaller than 2)? HOOTmag (talk) 23:19, 10 December 2012 (UTC)

You said "the X-axis", not "x≥2". It seems that you're correct; if a>b≥2 are integers, then the rhs dominates at x=2, so there is exactly one real root ≥2. However, it's not true that you can take a≥b without loss of generality. For example, if a=6, b=9, then there are solutions 2<x<2.1, and 3.5<x<3.6 . — Arthur Rubin (talk) 01:22, 11 December 2012 (UTC)

.

"However, it's not true that you can take a>b without loss of generality". Yes, a>b is not the only possible case for having a solution; The other possible case is b>c. Hence, when I spoke about the case in which the equation has no solution, I had to clarify also the condition of c: In point of fact, for every $c>b>a\geq 2$ the equation $a^{b^{c}}=b^{c^{a}}$ has no solution, because for every real x>b>a≥2 the function a^bx-b^xa is positive - i.e. not cutting the X-axis, and that's why my search - for points at which the function a^bx-b^xa cuts the X axis - was aimed at a specific case, e.g. when a>b. Anyways, I couldn't have written that the function a^bx-b^xa may also cut the X-axis when x>b - because x the variable itself - which is not known yet to be a point at which the function cuts the X-axis, and that's why I didn't mention the case of c>b.

.

Anyways, here is the proof for the absence of solutions in the case of $c>b>a\geq 3$ :

.

1. Every two real numbes $s,t$ satisfy $s+3^{t}>s$.

2. Every real $r$ satisfies $3^{r}>r$; So every real $r$ and every positive $t$ satisfy $t+3^{r}>r$.

3. Every two positive numbers $t>r$ satisfy $3^{t}-t>3^{r}-r$; So every three positive numbers $t>s>r$ satisfy $(3^{t}+s)-(3^{r}+t)>s-r$.

4. Every two positive numbers $s>r$, and every $x>r$, and every $y>s$, satisfying $y-x>s-r$, satisfy $3^{y}-3^{x}>s-r$, hence satisfy $r+3^{y}>s+3^{x}$.

5. Combine 1,2,3,4, by substituting $s+3^{t}$ for $y$, and $t+3^{r}$ for $x$, so it follows that every positive $t>s>r$ satisfy $r+3^{(s+3^{t})}>s+3^{(t+3^{r})}$, hence satisfy $3^{(3^{(r+3^{(s+3^{t})})})}>3^{(3^{(s+3^{(t+3^{r})})})}$, hence satisfy $(3^{3^{r}})^{((3^{3^{s}})^{(3^{3^{t}})})}>(3^{3^{s}})^{((3^{3^{t}})^{(3^{3^{r}})})}$, hence every $c>b>a\geq 3$ satisfy $a^{b^{c}}>b^{c^{a}}$. Q.E.D.

.

HOOTmag (talk) 13:08, 11 December 2012 (UTC)

Without checking, as I believe you're correct in full (although step 3 is wrong; it requires $r\geq \ln {\tfrac {1}{\ln 2}}$, it doesn't help the analysis. You made "c" a variable ("x"); excluding "c>b" is complicated, and is automatic in my analysis, anyway. — Arthur Rubin (talk) 14:13, 11 December 2012 (UTC)

If $a<b<c$, then;

1. $b>a>{\frac {a}{\ln b}}$, so we need only consider c = b
2. Substituting in my equation 3, moving terms from side to side, and factoring, we would need to verify

   $b(\ln b-\ln a)>\ln \ln b-\ln \ln a$

3. But

   ${\frac {\ln \ln b-\ln \ln a}{\ln b-\ln a}}<{\frac {1}{\ln a}}\leq {\frac {1}{\ln 2}}<2<b$

— Arthur Rubin (talk) 14:30, 11 December 2012 (UTC)

• As for my step 3: yes, so I've just replaced 2 by 3.
• As for your comment that "it doesn't help the avalysis": It really doesn't help your analysis, but it still does help mine.
• c in the equation $a^{b^{c}}=b^{c^{a}}$ is really a variable in the function a^bx-b^xa. I didn't understand what's wrong with that.
• As for excluding c>b: I've explained in my previous response why I'd excluded this case. However, I have already indicated (ibid.) that the case of a>b really does not exhaust the original question.

As for your analysis: yes, it's much more simple, and refers to a more general case, i.e. $a<b<c$, whereas my analysis only refers to the case of $c>b>a\geq 3$. However, I preferred an analysis that avoids infinitesimal considerations (e.g. derivatives, natural logarithms, and the like). HOOTmag (talk) 16:22, 11 December 2012 (UTC)

Unknown pattern

Can you find any pattern between this set of inputs and outputs? (4, 975), (0, 287), (1, 246), (2, 558), (3, 870), (4, 829), (5, 141), (6, 453), (7, 412), (8, 724), (9, 161), (7, 473), (8, 785), (9, 744), (0, 809). The reason there are multiple outputs for the same input is due to an ambiguity: the inputs are given modulo 10, so I don't know if the actual value is 1, 11, 21, etc. 149.169.54.0 (talk) 18:24, 5 December 2012 (UTC)

In that case, those aren't really inputs and outputs, but rather two different outputs, where only the last digit of the input is known. StuRat (talk) 18:43, 5 December 2012 (UTC)

Plotting the numbers in a scattergraph reveals no obvious pattern. Do you have any more information about the inputs and outputs that might help us? Robinh (talk) 19:17, 5 December 2012 (UTC)

Plotting $ye^{2\pi ix/10}$ revealed no pattern either. Bo Jacoby (talk) 09:58, 7 December 2012 (UTC).

Update on "Request for Help Deriving the Formula Describing the Probability that a Set of Event Outcomes Might Occur for Make-Up Homework"

     Hello again, everybody; I just wanted to let you all know that I replied to the thread that I created with some clarification that might help you guys answer my question:

     Maybe it would help you or whoever else is interested in helping me with this problem if I told you that I'm trying to write another function which restricts my original function, which I now clarify below through a few slight modifications which change the domain from the power set ${\mathcal {P}}\left(S\right)$ of the sample space $S$ to the set $\mathbb {R}$ of all real numbers and remove the sample space $S$ from the mapping…:

${\begin{aligned}P\colon \mathbb {R} &\to \left[0,1\right]\\A&\mapsto {}^{\left|A\right|}/_{\left|S\right|}\end{aligned}}$

…, to the domain defined by the members of the sample space $S$ as explained here by the article on function domains, like so:

${\begin{aligned}\left.P\right|_{S}\colon S&\to \left[0,1\right]\\A&\mapsto {}^{\left|A\right|}/_{\left|S\right|},A\in {\mathcal {P}}(S)\end{aligned}}$

The only problem is that, since this is a high-school Probability and Statistics class for which I'm doing my homework, my teacher wouldn't expect me to know how to define functions in terms of maps; in fact, I had never seen the map-based way of defining a function before I read about in the article on functions. That's why I wanted to know if could define a probability function restricted to the sample space $S$ of relevance to each problem in my homework that takes an event $A$ (or $E$, if you're only using a single event; each of my problems wants me to solve for multiple probabilities $P\left(A\right),P\left(B\right),\ldots$ in the context of a probability distribution) as its only argument while still allowing me to use this restricted domain to calculate its result ${}^{\left|A\right|}/{}_{\left|S\right|}$ in standard, $f(x)$ notation. So, what do I do?

— BCG999 (talk) 18:54, 5 December 2012 (UTC), Request for Help Deriving the Formula Describing the Probability that a Set of Event Outcomes Might Occur for Make-Up Homework, November 28^th, 2012

If this helps any, please let me know. BCG999 (talk) 18:54, 5 December 2012 (UTC)

     I just realized that I made a mistake in my last post because the first equation…:

${\begin{aligned}P\colon \mathbb {R} &\to \left[0,1\right]\\A&\mapsto {}^{\left|A\right|}/_{\left|S\right|}\end{aligned}}$

…should actually read as follows:

${\begin{aligned}P\colon \mathbb {R} &\to \left[0,1\right]\\A&\mapsto {}^{\left|A\right|}/_{\left|\mathbb {R} \right|}\end{aligned}}$

I'll correct it in the archived version of this post. Speaking of which, where is everybody who was looking at my original post? Please don't tell me that you've given up on me! I really need your guys' help, and I'll answer any questions that you might have about any posts in either this thread or its archived version if you need me to clarify anything in them.

Thanks,

     BCG999 (talk) 19:03, 9 December 2012 (UTC)

     Hey, everyone; I just wanted to let you all know that I've added another post to my original thread that may actually be the solution to my problem. I've quoted it here:

     Wait a minute, I think I just figured this out myself! To verify my thinking, could I say that the following function:

${\begin{aligned}\left.P\right|_{S}\colon S&\to \left[0,1\right]\\A&\mapsto {}^{\left|A\right|}/_{\left|S\right|},A\in {\mathcal {P}}(S)\end{aligned}}$

…is actually equivalent to $\left.P\right|_{S}\left(A\right)={}^{\left|A\right|}/_{\left|S\right|}$ when the event $A$ is either a subset $A\subseteq S$ of the sample space $S$ or a member $A\in {\mathcal {P}}(S)$ of the power set ${\mathcal {P}}\left(S\right)$ of the sample space $S$?

Here's hoping I'm right,

     BCG999 (talk) 20:40, 9 December 2012 (UTC)

— BCG999 (talk), Wikipedia:Reference desk/Archives/Mathematics/2012 November 28#Request for Help Deriving the Formula Describing the Probability that a Set of Event Outcomes Might Occur for Make-Up Homework

Thanks in advance for responding if you do,

     BCG999 (talk) 20:58, 9 December 2012 (UTC)

Natural Numbers/Algebra

Given "n", a natural number.

- What number is the next consecutive natural number to "n"?

- Would the answer be "2" or "2n". I am debating about which one I should write as my answer. Because I also have a question that will help me solve: "Are natural numbers able to have algebraic letters (example- "2n")? But I don't think so. I think that the answer is most likely to be "2", correct. Because "2" is a "natural number" and "2n" is not. --RossSLynch (talk) 23:42, 5 December 2012 (UTC)

The natural numbers are the counting numbers 1, 2, 3, ... and so on. If n is a natural number, the next consecutive natural number is n+1 Dolphin (t) 00:30, 6 December 2012 (UTC)

Everybody forgets zero. « Aaron Rotenberg « Talk « 09:40, 6 December 2012 (UTC)

Natural number: "There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term designates the non-negative integers {0, 1, 2, 3, ...}. The former definition is the traditional one ..." - It's not so much "forgetting" as "explicitly excluding for not adhering to the definition they're using". -- 71.35.111.52 (talk) 17:49, 6 December 2012 (UTC)

Agreed, it's n+1. 2 and 2n are only the next consecutive natural number if n happens to be 1. If n is anything else, like 10, then only n+1 gives you the next consecutive natural number, 11. StuRat (talk) 18:02, 6 December 2012 (UTC)