Wikipedia:Reference desk/Archives/Mathematics/2008 December 16

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December 16

Category Theory question from Mac Lane's book

Today, I am working on a category theory homework. It is actually pretty simple I believe but I may be messing something up because there is an assumption given in the problem that I do not see as necessary. I think I can do the problem without it.

Problem IV.3.3

If $\langle G,F,\varphi \rangle :X\rightharpoonup A$ is an adjunction with G full and every unit $\eta _{x}$ a monic, then every $\eta _{x}$ is also epi.

I have a theorem from section IV.3 that says

For an adjunction $\langle G,F,\eta ,\epsilon \rangle :X\rightharpoonup A$: (i) G is faithful if and only if every component $\epsilon _{a}$ of the counit $\epsilon$ is epi, (ii) G is full if and only if every $\epsilon _{a}$ is a split monic. Hence G is full and faithful if and only if each $\epsilon _{a}$ is an isomorphism $FGa\cong a$.

So, I form the dual of the statement, which is

If $\langle F,G,\varphi ^{-1}\rangle :A\rightharpoonup X$ is an adjunction with G full and every counit $\eta _{x}$ epi, then every $\eta _{x}$ is a monic.

Okay, so now I can use the (ii) of the theorem and it tells me right away that G is full if and only if $\eta _{x}$ is a split monic since $\eta _{x}$ is the counit now. Thus, it is split monic, which is more than just being monic, isn't it? Monic means cancellable. Split monic means even more that there exists an inverse, which also means it is cancellable. So, I'm done proving the dual, so the original is also correct. But, the problem is I never used that $\eta _{x}$ is epi in the dual statement. So, why do they tell me that $\eta _{x}$ is monic in the original? Usually, this means I messed up! Thanks for any help. StatisticsMan (talk) 03:23, 16 December 2008 (UTC)

I'm a bit confused by the way you formed the dual statement. I suspect that a correct dual statement would have F full rather than G.

Here's a solution, though:

We want to show η_x: x → GFx is epi, so let f,g: GFx → x′ be such that fη_x = gη_x. It's enough to show f = g. Since G is full, there are p, q: Fx → Fx′ such that Gp = η_x′f, Gq = η_xg. Then the adjoints of p and q, respectively (Gp)η_x = η_x′fη_x and (Gq)η_x = η_x′gη_x, are equal. Therefore p and q are themselves equal. Thus η_x′f = Gp = Gq = η_x′g. Since η_x′ is monic, this proves f = g. 67.150.254.50 (talk) 14:11, 16 December 2008 (UTC)

You could certainly be right about the dual being incorrect. But, I used this table in that section of Mac Lane's book and it says the dual of saying F is full is that F is full, the same functor. Any way, thanks for the help. I'll probably just talk to my professor today any way. StatisticsMan (talk) 15:37, 16 December 2008 (UTC)

If you have a functor F:A → B, then certainly, if you view it as a functor A^op → B^op, it is full one way if and only if it is full the other way. But in your statement, you appear to be replacing A with X^op and X with A^op. Therefore, you're really exchanging the roles of F and G. 67.150.252.175 (talk) 01:44, 17 December 2008 (UTC)

How many subtractions do you need to obtain number 32, using the numbers 1, 5, 6, 7 and 9?

This is not a homework question as I am not a student, but I am having the darndest time trying to answer this. If you don't want to answer it for for me, fine, but how would I start by solving it? I don't know where to begin...--Emyn ned (talk) 19:15, 16 December 2008 (UTC)

I don't understand the problem. You can't do it just using subtraction, since all of the numbers are positive and less than 32 (unless you're allowed to put brackets in strange places, perhaps, but that would really just be a combination of subtraction and addition). What operations are you allowed to use? Are you just allowed to use those numbers, or also numbers formed using those as digits (eg. can you combine 1 and 5 to make 15?)? --Tango (talk) 19:26, 16 December 2008 (UTC)

Oh, and are you allowed to use each number more than once? And do you have to use all the numbers? --Tango (talk) 19:27, 16 December 2008 (UTC)

Isn't this trivial? Note that:

51 - 19 = 32

so the answer is one subtraction. In general, with problems like this one, aim for for the smallest numbers first. Topology Expert (talk) 19:30, 16 December 2008 (UTC)

I dont know the answer to you questions. I am taking an online IQ quiz and this is listed as one of the questions with no further instructions. http://www.cerebrals.org/index.php?go=tests --Emyn ned (talk) 19:33, 16 December 2008 (UTC)

Is what you wrote in the header the exact question, word for word? If so, the test is rubbish, just ignore it. In fact, ignore online IQ tests in general - they generally give inflated results in order to encourage you to pay a large sum of money for an automatically generated "report" saying how clever you are in lots of technical sounding but meaningless detail. In fact, I'll go one step further than that - ignore IQ's, they are a measure of how good you are an IQ tests, nothing more (there is a correlation between that and intelligence, certainly, but it's not a very strong one). --Tango (talk) 19:36, 16 December 2008 (UTC)

Actually, I am trying to challenge myself as I already consider myself quite intelligent (despite my inability to type correctly). As intelligent as I am, I was greatly humbled by this question....--Emyn ned (talk) 19:39, 16 December 2008 (UTC)

The number 32 can be obtained from the given integers using no fewer than five subtractions:

$32=1-(1-7-9-9-9)$

$32=1-(1-5-9-9-9)$

Cheers, siℓℓy rabbit (talk) 19:54, 16 December 2008 (UTC)

That's 34. Algebraist 19:57, 16 December 2008 (UTC)

Apparently you didn't account for the parentheses. Subtraction is, of course, nonassociative. siℓℓy rabbit (talk) 20:43, 16 December 2008 (UTC)

1-(1-7-9-9-9)=1-(-33)=34, no? Algebraist 20:49, 16 December 2008 (UTC)

It certainly did when I was at school... --Tango (talk) 20:56, 16 December 2008 (UTC)

No, that is 32 (at least I think it is); add it up using a calculator, perhaps.

I agree with Tango: IQ tests are rubbish (they test 'general understanding' rather than understanding in a particular field). The only IQ test that is at least a bit decent is this. Especially the mathematics section is general rubbish (as you may find out if you read it; it just contains what you may find out by memorizing mathematics) but does include concepts such as fibre bundles! Topology Expert (talk) 20:01, 16 December 2008 (UTC)

Dang! Here's a similar one: How many additions do you need to obtain number 10, using the numbers 1, 3, 3 and 5? I have no idea where to begin...--Emyn ned (talk) 20:00, 16 December 2008 (UTC)

5+5 = 10 (please stop asking questions here unless you have actually thought about them). Topology Expert (talk) 20:02, 16 December 2008 (UTC)

Since 3 was listed twice I think it's fair to assume you can't use numbers more than once (unless they are listed more than once). In that case, there is no way of doing it using just addition, if you are allowed other elementary operations, then you can do it without any additions: 5*(3-1). There are really stupid questions - I suggest you find a better way to challenge yourself. --Tango (talk) 20:55, 16 December 2008 (UTC)

How about: 32 = 6 - (1 - 9 - 9 - 9) = 6 - (-26)?. -- Tcncv (talk) 01:43, 17 December 2008 (UTC)

Emyn ned: as Tango suggests, the interesting aspect is the philological one, that is, guess what was the original statement of the problem you are quoting. I conjecture: you can only rearrange the symbols "2 3 3 5" and a minimal number of "+", possibly one, (and no other symbols) to get 10. Yes, let's stop.--PMajer (talk) 14:24, 17 December 2008 (UTC)

I GOT IT!!!! THE ANSWER IS 57 - 19 - 6 = 32!! So, it's 3 Subtractions! YAY! --Emyn ned (talk) 15:30, 17 December 2008 (UTC)