April 30

Hi all, as you know, I've been studying for a PhD qualifying examination in Analysis. I am having trouble finding a good (large) store of exercises having to do with the theorems of Fubini and Tonelli. I imagine that many of you have encountered things like this in the past, and I was hoping you would share your sources with me. Online, in a book, whatever. I'm all about it. Thanks in advance! –King Bee ^{(τ • γ)} 13:48, 30 April 2007 (UTC)[reply]

Berkeley Problems in Mathematics is a good book a friend of mine bought to study for exams. Beside that, there are tons of problems online if you just search for them. Here's a web page which contains many links, though some are broken. But, to find these, just go to any major college's math department page and look around. Or, just search for "math qualifying exams" or something like that. http://www.math.purdue.edu/~bell/Quals/ StatisticsMan 06:36, 1 May 2007 (UTC)[reply]

Ahh, thanks for the advice! –King Bee ^{(τ • γ)} 14:02, 1 May 2007 (UTC)[reply]

Is there another answer for difference as in a subtraction problem? —The preceding unsigned comment was added by 70.254.42.11 (talk) 21:24, 30 April 2007 (UTC).[reply]

I'm not sure I understand your question. If you mean 'can "the difference between ${\displaystyle a}$ and ${\displaystyle b}$" mean something other than ${\displaystyle |a-b|}$', then I would say probably not, as long as ${\displaystyle a}$ and ${\displaystyle b}$ are numbers. Algebraist 21:34, 30 April 2007 (UTC)[reply]

I think he/she meant:

if a - b = c

Then can there be a different answer where

a - b = d  and    d is not equal to c

In other words "Can there be more than one solution for a subtraction problem?" 202.168.50.40 22:14, 30 April 2007 (UTC)[reply]

If that is the intended meaning, then we have the following argument:

${\displaystyle {\begin{aligned}a-b&=c&\qquad &{\text{(first answer)))\\a-b&=d&\qquad &{\text{(second answer)))\\(a-b)-(a-b)&=c-d&\qquad &{\text{(subtract on both sides)))\\0&=c-d&\qquad &{\text{(difference of equals on left)))\\d&=c&\qquad &{\text{(add ))d{\text{ to both sides)))\end{aligned))}$

It is interesting to compare this to ±√2, solutions to x² = 2, where we can have two different answers. --KSmrq^T 23:09, 30 April 2007 (UTC)[reply]

What use is that?

Logical inclusive OR

• 0+0=0
• 0+1=1
• 1+1=1
• 1+1=1

Logical AND

• 0×0=0
• 0×1=0
• 1×0=0
• 1×1=1

But consider...

• 1+1=2

Logical _____?

--Kirby♥time 23:59, 30 April 2007 (UTC)[reply]

"1+1=2" isn't logical anything. It is integer arithmetic. It is useful, in part, because nonnegative integers are those entities that are used to count things. That is, they answer "How many ..." questions. (I imagine you knew all that, by the way. If I have not answered what you asked, perhaps you could clarify your question?) Tugbug 00:45, 1 May 2007 (UTC)[reply]

This mathematical operation is called "addition". If you need to be more specific, you can say it is addition of natural numbers. See also Wikipedia:Reference desk/Archives/Mathematics/2007 April 18#Why do 1+1=2?.  --Lambiam^Talk 05:44, 1 May 2007 (UTC)[reply]

You are looking at three different abstract algebras. The integers modulo 2 — denoted by Z₂, Z/2Z, or Z/2 — have only two elements, 0 and 1; this ring is isomorphic to a Boolean algebra, giving the interpretation as logic you imply. The natural numbers — denoted by N — are a different algebra, and we need to choose a way to map to logic. We could, of course, use the homomorphism from N to Z₂ whose kernel is the even numbers, then compose; this would imply that addition maps to exclusive OR. But we could also map zero to FALSE and nonzero to TRUE; then n+m delivers a nonzero (TRUE) result only if one of the addends is nonzero (TRUE), which is logical inclusive OR. This is not fanciful; the programming language C takes exactly this latter approach. --KSmrq^T 06:24, 1 May 2007 (UTC)[reply]