Wikipedia:Reference desk/Archives/Mathematics/2007 October 13

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October 13

okay, so is { 42, {} } a subset of { 42 }?

Is { 42, {} } a subset of { 42 }? Why or why not? Is {} a subset of {42}? Why or why not?

And if only the latter but not the former what explains the difference? —Preceding unsigned comment added by 84.0.126.201 (talk) 10:34, 13 October 2007 (UTC)

• The integer 42 is a member of the set {42}.
• The set {42} is a subset of the set {42} because every set is a subset of itself.
• The empty set {} is also a subset of {42} because the empty set is a subset of every set (more formally, this is because the empty set does not contain an element that is not a member of {42}).
• {42, {}} is not a subset of {42} because it contains an element, {}, which is not a member of {42} - {} is a subset of {42}, but not a member of {}.
• However, {42} is a subset of {42, {}}. Gandalf61 10:49, 13 October 2007 (UTC)

I get most of your statements except the fourth one: "{42, {}} is not a subset of {42} because it contains an element, {}, which is not a member of {42} - {} is a subset of {42}, but not a member of {}". If the empty set is a subset of {42}, why can't I add it to 42, which is a member, and get a new subset based on a previous subset + one member? I'm not counting anything twice... —Preceding unsigned comment added by 84.0.126.201 (talk) 11:09, 13 October 2007 (UTC)

By add it, I mean make a new subset consisting of a given subset plus a member of the original set that isn't in it? I mean, if you tell me {apple, banana} is a subset of Whatever, and that pineapple is also a member of Whatever, isn't that enough for me to construct the subset {apple, banana, pineapple} and be sure it really is a subset of Whatever? Likewise, if you tell me {} is a subset of Whatever, and that pineapplei s also a member of Whatever, then isn't that enough for me to construct the subset {{}, pineapple} and be sure it really is a subset of Whatever? —Preceding unsigned comment added by 84.0.126.201 (talk) 11:11, 13 October 2007 (UTC)

Note the differences between "member" and "subset", and between {} and {{}}. {} is the empty set, the set containing 0 elements. {{}} is a set containing a single element, that element being the set {}. So, if {{}} is a subset of whatever, and pineapple is an element of whatever, then {{},pineapple} (the set containing 2 elements, {} and pineapple) is a subset of whatever. But that's not what you have here. You have that {} is a subset of {42}, not that {{}} is a subset of {42} (indeed, {{}} is not a subset of 42 because {} is in {{}} but not in {42}). On the other hand, if you take {} (a subset of {42}) and add to it 42 (an element of {42}) then you get {42} which is indeed a subset of {42}. -- Meni Rosenfeld (talk) 11:17, 13 October 2007 (UTC)

... wait a minute, I think I get it. Is {{42}} a set, but not a subset of {42}, since it contains an element, {42}, which is not in the latter? Likewise, is { {} } a set, but not a subset of {42}, since it contains a member, {}, which is not a member of the original set? So then are all of these sets:

• {}
• {{}}
• {42}
• {{42}}
• { {{42}}, {}}
• { {{42}}, {{{}}} }
• { {},{{}},{{{}}},{{{{}}}},42,{42},{42,{}},{42,{{}}},{{42}},{{42,{}}},{{42,{{}}}},{{{42,{{}}}}} }

????

Isn't that pretty stupid though? What keeps someone from making a set like this:

• { {}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, {{{{{{}}}}}}, {{{{{{{}}}}}}}, {{{{{{{{}}}}}}}} }

???? Thank you.

Not stupid at all - you have discovered a method of constructing the natural numbers. Gandalf61 11:32, 13 October 2007 (UTC)

Yep, and of course, you got everything right.

It goes even deeper than that. It is accepted that sets are very important in mathematics, and that we would do well to have a system that incorporates them. There is then the question, do we need anything other than sets? The answer is more or less "not at all!". You can do pretty much everything with just sets and nothing else. But we then have the question, "if there are only sets, what will be the elements of the sets"? Here the empty set comes to our rescue - having no elements, we can construct it without having any elements yet. But now that we have it, we can use it as an element and construct the set {{}}. And now we know 2 sets, and we can do all sorts of crazy things, such as {{{}}} and {{},{{}}}. We can then continue constructing richer sets. We need additional tools if we want to be able to construct, say, infinite sets (that's basically what ZFC does, giving us tools to construct sets from the ground up). And we can then define all known mathematical concepts as particular sets, like the construction of natural numbers Gandalf has mentioned - from which we can move on to construct rational numbers, real numbers, functions, and so on. -- Meni Rosenfeld (talk) 11:52, 13 October 2007 (UTC)

wow! how? even calculus? —Preceding unsigned comment added by 81.182.100.153 (talk) 16:44, 13 October 2007 (UTC)

Yes, of course. There's no problem, really; the most important concept in calculus is functions, and those are readily constructible as sets. -- Meni Rosenfeld (talk) 16:46, 13 October 2007 (UTC)

btw, how do you construct functions from sets? —Preceding unsigned comment added by 81.182.100.153 (talk) 22:22, 13 October 2007 (UTC)

See below. -- Meni Rosenfeld (talk) 22:32, 13 October 2007 (UTC)

Okay, I'm sold on all this "constructing all mathematical concepts as particular sets" -- so how do we do that? —Preceding unsigned comment added by 81.182.100.153 (talk) 17:01, 13 October 2007 (UTC)

We start at the beginning, and don't leave anything out... You can begin with defining the ordered pair (a,b) as the set {a,{a,b}}. Then a function is a set of ordered pairs satisfying a property (that no two pairs have the same first element). An ordinal is a transitive (under membership) set whose elements are also transitive. A natural number is a finite ordinal. Addition and multiplication can be defined using cardinalities. Cartesian products are defined normally. An integer is an equivalence class of $\mathbb {N} \times \mathbb {N}$ under the equivalence $(a,b)\equiv (c,d)\iff a+d=b+c$. A rational number is an equivalence class of $\mathbb {Z} \times \mathbb {Z} ^{*}$ under the equivalence $(a,b)\equiv (c,d)\iff ad=bc$. A sequence of rational numbers is a function $f:\mathbb {N} \to \mathbb {Q}$, and equivalent Cauchy sequences can be defined. A real number is an equivalence class of Cauchy sequnces. The rest follows in the same way (you can define most objects in terms of functions or equivalence classes).

Of course, there are a lot of theorems to be proven along the way. -- Meni Rosenfeld (talk) 17:16, 13 October 2007 (UTC)

For some idea of exactly how many, see Principia Mathematica (and, more generally, implementation of mathematics in set theory). —Ilmari Karonen (talk) 18:29, 13 October 2007 (UTC)

"zfc" second axiom

I'm getting confused on the second axiom given on the page ZFC - the text says as I read it that for a set x there is a member y that is disjoint - which I think means that y has no members that is in x. But this seems to contradict y being a member of set x.

Am I misreading this or what? A simple explanantion would be best - just a simple example please thanks. (the other axioms haven't bothered me.)83.100.254.51 13:27, 13 October 2007 (UTC)

I think you're probably getting confused by the difference between set membership and subsethood. Consider, as an example, the set A={$\emptyset$}. Then A has an element, $\emptyset$, which has no elements at all, and thus no elements in common with A, so $\emptyset$ is disjoint from A. Algebraist 13:35, 13 October 2007 (UTC)

Another example: If $x=\{\{1\},\{2\}\}$, then $y=\{1\}$ is a witness; $y\in x$ but y and x are disjoint - the element of y is 1 and the elements of x are {1} and {2}. Remember, 1 is not the same as {1}. -- Meni Rosenfeld (talk) 13:43, 13 October 2007 (UTC)

(as long as you remember not to define 1 to be {2} when inventing the natural numbers) Algebraist 13:46, 13 October 2007 (UTC)

Well, yeah, since we're speaking ZFC I'm also assuming the standard construction 1 = {{}}, 2 = {{},{{}}}. -- Meni Rosenfeld (talk) 13:48, 13 October 2007 (UTC)

I understood the two examples..

In 'zfc' can a set X contain (1 2 3 4 5) ie just those five numbers?

If so what would be the member(s) Y that is/are disjoint?83.100.254.51 13:51, 13 October 2007 (UTC)

Yes, ZFC guarantees the existence of the set {1, 2, 3, 4, 5} (assuming you have defined those numbers of course). According to the standard construction, 1 = {0}, so since $0\notin \{1,2,3,4,5\}$, you can take $y=1$. -- Meni Rosenfeld (talk) 13:54, 13 October 2007 (UTC)

The point of my joking aside to Meni earlier is that in ZFC we can only talk about sets, so if we want to talk about numbers, we have to define them to be certain sets. We're using the standard definition. Algebraist 14:06, 13 October 2007 (UTC)

I'm a little bemused. I understand using the constructs for natural numbers, but for consistency shouldn't y=1 be replaced therefor with y={{}} - it seems inconsistant to replace the numerals with their abstractions in one set but not to do so in the other?

mmh - is there are notation for "not a set" ? eg (in simple set theory) if X is the set that doesn't contain itself or it's members (a simple logical contradiction)

(also is there a notation for 'not a number' eg using x+1=2, x+2=4 then x is not a number - i suppose x could be an empty set)83.100.254.51 14:40, 13 October 2007 (UTC)

The idea is that once we have defined 1 = {{}}, then 1 and {{}} are the same thing. So {1} and {{{}}} are the same set, and the set {1,{{}}} contains one element. Our definitions should be kept in mind when making statements.

In ZFC there is no such thing as something which is not a set. Don't be confused with the role of contradictions. It's not like "a set which is not a subset of itself exists, but we shouldn't talk about it or our teacher will get angry". Such a thing simply doesn't exist.

Same for your third paragraph, there simply does not exist any x such that x+1=2 and x+2=4. It's not like "the x which satisfies this is naughty", it's just that there is no such x. The same way that I can't say "my younger sibling is not a person". I don't have a younger sibling, the phrase "Meni Rosenfeld's younger sibling" is just a meaningless string of symbols, and as it means nothing, I can't discuss whether whatever is meant by it is a person or not. Compare the phrase "Meni Rosenfeld's brother", which is a meaningful phrase refering to a particular person, and I can discuss the age, eye color and occupation of that person.

Note that there are deeper issues of sense and reference involved; I am speaking about references here, as is usual in mathematics. -- Meni Rosenfeld (talk) 15:22, 13 October 2007 (UTC)

Approximations of trig functions

I think most people are familiar with the 45-45-90 and the 30-60-90 triangles, but say we want to approximate the tangent of an angle with a magnitude of, say, 47 degrees. Is there a simple way to do it other than using the Taylor Series? Pandacomics 14:12, 13 October 2007 (UTC)

how about sin (45+2) = sin45cos2 + cos45sin2 (do the same for cos) - you've already got sin 45, and sin/cos 2 will rapidly converge using the taylor type series - but this time it converges much quicker - (maybe you already were doing that?)83.100.254.51 14:44, 13 October 2007 (UTC)

Also see Sine#Computation - which gives other methods - though maybe not so simple - if it's for a computer CORDIC may be interesting but not as simple.83.100.254.51 14:51, 13 October 2007 (UTC)

This (might be) right

tan(a+b) = ( tan a cos b + sin b )/(cos b - tan a sin b)

In you case using a look up table for major values of tan would help, but also tan 45 = 1 so tan (47) = tan (45+2) = ( cos 2 + sin 2 )/(cos 2 - sin 2 ) - you still need to convert degrees to radians which is a pain - but once done the series will converge very rapidly. —Preceding unsigned comment added by 83.100.254.51 (talk) 15:31, 13 October 2007 (UTC)

A good way of calculating the Sine is: If x is lesser in magnitude than some threshold, then $\sin x=x$; otherwise, $\sin x=\sin {\tfrac {x}{3}}(3-4\sin ^{2}{\tfrac {x}{3}})$. This scales well to large values of x without any additional effort. -- Meni Rosenfeld (talk) 15:37, 13 October 2007 (UTC)

Anyway, what's wrong with using the Taylor series? -- Meni Rosenfeld (talk) 16:34, 13 October 2007 (UTC)

Sorry, I meant figuring out trig approximations without a calculator or tables (as well as no Taylor, cause it takes a long time to number crunch). The sum thing is convenient yeah, but only if you have tables/calculators handy. The Taylor series is just tedious. Simplistic, but time-consuming. Pandacomics 00:13, 14 October 2007 (UTC)

You mean like they did "back in the day" when we were still writing the sine tables ?

(I'll exclude measurement based devices - eg a very round circle - a length of string and a rule ... actually this is a pretty good way..)

I do know of another method - it gets a sine to a n^th iteration of accuracy - unfortunately for each n you have to recalculate the whole thing.. here goes..

Take the angle to mean a distance around a circle of radius 1. to the nth degree of accuracy assume that that length (the arc) is the same as 2ⁿ chords... there for the length of a chord would be angle/2ⁿ - fairly simple maths..

(step A) Then find the length of the chord connecting 2 chords (this is simple 'pythag/eucli' stuff and will assume unless you ask that you can get this) - it involves square root (clue)

repeat step A a further n-1 times - this will give you the length of the chord connecting the extremities of the arc.

You now have the chord length - using simple pythagorean contructions (another square root at least) - you can get the sin and cosine.

If you really enjoy calculating square roots you will love this.! I can't think of any other methods of hand.87.102.19.106 03:54, 14 October 2007 (UTC)

See also Generating_trigonometric_tables section "A quick, but inaccurate, approximation" might be useful as well. And thanks for asking, this question has a special hole in my heart.87.102.19.106 03:58, 14 October 2007 (UTC)

I still recommend the tailor series of sine and cos if you want to do the calculation by hand. It's really the fastest method. For 47 degrees, do it as recommended above. First, calculate the sine and cosine of 2 degrees: sin(2 deg) = (pi/90)(1 - (pi/90)^2/6), cos(2 deg) = 1 - (pi/90)^2/2 should be a fairly good approximation (because the fourth order terms are small) and aren't too difficult to calculate either: you just have to memorize the value of pi, and it's only a few multiplications from there. From then, you get lucky because sin(45 deg) = cos(45 deg) cancels out: tan(47 deg) = (sin(45 deg)cos(2 deg) + cos(45 deg)sin(2 deg))/(cos(45 deg)cos(2 deg) - sin(45 deg)sin(2 deg)) = (cos(2 deg) + sin(2 deg))/(cos(2 deg) - sin(2 deg)). – b_jonas 09:29, 14 October 2007 (UTC)

Btw, if you really want to do computations by hand, it could be handy to print small tables of all functions you need. For example, print sine and cosine for four digits at every 0.1 degrees (between 0 and 45 degrees) and use 1st derivatives from there to calculate the sine of an angle given to two decimals in degrees to four decimal places. – b_jonas 09:35, 14 October 2007 (UTC)

Of course, in all cases we are assuming you have no calculator or table, as otherwise the problem would be trivial. But there is no such thing as a free lunch. If you want a (good) approximation for a trig function, you have to put some effort in it. True, some meals are cheaper than others, but a clever usage of Taylor series (as described by various people above) may be the easiest way. You can also consider the method I described above (and similarly for Cosine, $\cos x=1-{\tfrac {x^{2}}{2}}$ for $x\approx 0$ and $\cos x=\cos {\tfrac {x}{3}}(4\cos ^{2}{\tfrac {x}{3}}-3)$ otherwise), which doesn't require any sort of cleverness but might require more calculations for a given accuracy. -- Meni Rosenfeld (talk) 11:46, 14 October 2007 (UTC)

Or, subtract the nearest multiple of $\pi /2$ to obtain $-\pi /4\leq x\leq \pi /4$, then use two terms of the Taylor series for cos or sin for a maximum absolute error less than 0.02 (three terms gives an error less than 0.0004). The only catch is that you have to keep track of signs and when to use the sine/cosine series, but if you're like me and easily mess things like that up, you write down a table like this

-pi/4 < x < pi/4  : sin(x) = sin(x), cos(x) = cos(x)

pi/4 < x < 3*pi/4 : sin(x) = cos(x-pi/2), cos(x) = -sin(x-pi/2)

3*pi/4 < x < 5*pi/4 : sin(x) = -sin(x-pi), cos(x) = -cos(x-pi)

5*pi/4 < x < 7*pi/4 : sin(x) = -cos(x-3*pi/2), cos(x) = sin(x-3*pi/2)

and maybe also compute the decimal values of the first few multiples of $\pi /4$ in advance for easy reference. For larger (or negative) x, start by subtracting (adding) the nearest multiple of $2\pi$. Fredrik Johansson 13:20, 14 October 2007 (UTC)

Yet another alternative for calculating sines and cosines in general would be to use the taylor series for only sin(1) and cos(1) to a high degree of accuracy - then use the formulae for sin(x/2) and cos(x/2) ro get sin(0.5),sin(0.25),sin(0.125) .. sin (2^-n) - then when you want a sin - write your angles as binary and used the sin(a+b+...) formula to get the value - in other words generate a table of sines and cosine that has maximum usefulness for the work you need to do to make it.87.102.82.26 18:05, 14 October 2007 (UTC)

I stumbled on this http://mathworld.wolfram.com/WallisFormula.html - I'm sure there are other expressions giving sine - but I'm not familiar with them, see also Wallis product Infinite_product#Product representations of functions for some other variations - I don't actually undertand these myself (as yet) but might be useful to you.87.102.82.26 18:23, 14 October 2007 (UTC)

The Wallis formula is of theoretic interest, it is unsuitable for actual calculation of the sine. -- Meni Rosenfeld (talk) 20:30, 14 October 2007 (UTC)

squaring a circle

Would someone please explain me the way E.W.Hobson constructed a square equalling the area of a given circle? (i.e., the construction of approximated —Preceding unsigned comment added by Kasiraoj (talk • contribs) 15:04, 13 October 2007 (UTC)

        would someone explain how E.W.Hobson constructed a square equaling its area to the area 
        of a given circle or the method he followed to construct a line segment of length 
        squareroot of pi Kasiraoj 15:36, 13 October 2007 (UTC)

The method he used was only approximate - I can't find an actual description but it looks like the book you would need is " E. W. Hobson, Squaring the Circle, Chelsea, 1913" - to get this ask at your local library - it would probably require an interlibrary loan83.100.254.51 16:19, 13 October 2007 (UTC)

Constructing Regular Polygons

I have long wondered if it is possible to construct, say a Regular Pentagon, using only a compass and straight-edge. Is this possible? If so, how? A math-wiki 23:16, 13 October 2007 (UTC)

Constructible polygon. There ya go. It actually has an animation for the regular pentagon. Gauss found a contruction of a 17-gon at age 19. The article also gives the condition must be met for an n-gon to be constructible: A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes. risk 00:15, 14 October 2007 (UTC)

The animated procedure on that page is unnecessarily cumbersome: once the first pentagon edge has been constructed, it should of course simply be copied along the circle using the compass. The author of the animation was apparently tipped off about this more than a year ago and hasn't responded. Does anyone here, who knows how to, feel like fixing it? --mglg_(talk) 00:14, 15 October 2007 (UTC)

Actually, according to the article, Gauss has proven a construction is possible but has not found an actual construction himself. -- Meni Rosenfeld (talk) 18:18, 14 October 2007 (UTC)

Though the article does assert that he had an expression for cos(2pi/17) involving only algebra and square roots, from which a (monstrously inefficient) construction is trivial. I don't know for certain, but it seems very likely that this was also trivial for Gauss. Algebraist 21:08, 14 October 2007 (UTC)

You are absolutely correct, I missed that. According to Heptadecagon, a 64 step construction was found some years later, and according to Heptadecagon at MathWorld Richmond found a 25 step solution in 1893. I'm sure my high school math book said Gauss found a construction, rather than proving it's existence. I guess they were wrong. risk 21:28, 14 October 2007 (UTC)

Well, our article could be wrong, too... I'm not sure about any details, but it is quite possible that he knowingly had a way to extract a theoretically possible construction, but it was too long for any human to actually construct it. Erchinger would then be the first to provide a tractable construction. -- Meni Rosenfeld (talk) 21:33, 14 October 2007 (UTC)

My High School Geometry text also said Gauss constructed it. A math-wiki 23:57, 14 October 2007 (UTC)

Please provide a citation( title, author, ISBN for this book, Although a lot of math books contain typos and other weird stuff, its extrodinarly uncommon to make a historical mistake. I will look fro refrences in the biography of CF Gauss, where he proved it was contructable, but didnt provide the methoud. ( and btw, Enreste Galois worked on this too )

Just wondering, has anyone come up with a way to know what is the minimum number of steps needed to construction and n-gon, provided it can be constructed? A math-wiki 00:04, 15 October 2007 (UTC)

First, I'm impressed that your books even mention this - back in my day, we were never taught any of those interesting things. I wouldn't be too surprised, though, if they mixed up a proof of possibility with an actual (carried out) construction - that is the nature of high school texts. -- Meni Rosenfeld (talk) 00:59, 15 October 2007 (UTC)

It was elucidated in G.E.B. By Douglas Hofstatler. Basically, you need to find as many construction methods, and find out what they cover. ( I was just reading about a construction method for cronstructing root^3(2) that was rigorus. At frist, you make need to start with a sieve, there may be a smallest n-gon that is contructable, but has no know method to contruct it.