 Infinity and Jelly Doughnuts davegage Posers and Puzzles 21 May '05 09:39
1. 21 May '05 09:392 edits
Which of the following sets contains the largest number of elements? Which of the following sets contains the smallest number of elements?
Rank the sets in order of number of elements from smallest to largest.

0. The set of all positive whole numbers and 0: {0,1,2,...}
1. The set of all positive whole numbers: {1,2,3,...}
2. The set of all positive multiples of 10: {10,20,30,...}
3. The set containing all numbers on the number line between 0 and 1: (0,1).
4. The set containing all numbers on the number line between 0 and 1, plus also 0 and 1: [0,1].
5. The set containing all numbers on the number line between 0 and 1, plus also 0 but not 1: [0,1).
6. The set containing all numbers on the number line between 0 and 2: (0,2).
7. The set containing all non-negative numbers on the number line: [0,+ Infinity).

For extra credit: Without looking it up on the internet, where does the expression 'Infinity and Jelly Doughnuts' come from?
2. 21 May '05 11:112 edits
the one to infinity is the largest or the positive whole numbers... too obvious though
3. 21 May '05 11:37
Originally posted by davegage
Which of the following sets contains the largest number of elements? Which of the following sets contains the smallest number of elements?
Rank the sets in order of number of elements from smallest to largest.

0. The set of all positive whole numbers and 0: {0,1,2,...}
1. The set of all positive whole numbers: {1,2,3,...}
2. The set of all positive m ...[text shortened]... ing it up on the internet, where does the expression 'Infinity and Jelly Doughnuts' come from?
As the answer is obvious to any mathmo, I won't give an answer, so that others can ponder it first. Instead, I'll ask another question:

The sets {1,2,3,4,5,...,infinity} and {-infinity,1,2,3,4,5,...} can clearly be put in 1-1 correspondence. But in what sense is the first set 'bigger' than the second?
4. 21 May '05 15:021 edit
Originally posted by Acolyte
As the answer is obvious to any mathmo, I won't give an answer, so that others can ponder it first. Instead, I'll ask another question:

The sets {1,2,3,4,5,...,infinity} and {-infinity,1,2,3,4,5,...} can clearly be put in 1-1 correspo ...[text shortened]... e. But in what sense is the first set 'bigger' than the second?
well, when setting up this one to one corrispondance, one will note that the member of each pair belonging to the first set is larger, thus the first set it'self must theoreticaly sum to a larger number.

actualy i see a more obvious way of getting to the same place: for every number x in the second set, there is a number -x, thus the sum of the second set is 0. the sum of the first set is infinity.

edit:
someone explane "infinity and jelly doungust" (spellling?)
5. 21 May '05 16:16
well, when setting up this one to one corrispondance, one will note that the member of each pair belonging to the first set is larger, thus the first set it'self must theoreticaly sum to a larger number.

actualy i see a more obvious way of getting to the same place: for every number x in the second set, there is a number -x, thus the sum of the s ...[text shortened]... e first set is infinity.

edit:
someone explane "infinity and jelly doungust" (spellling?)
Possibly, except that summing {-infinity,1,2,3,...} is impossible in any normal sense.

Suppose all I care about is the order of the elements within a set, so {1,2,3} is the same as {4,10,357}. How does this make the two sets I gave before different, and it what way can I say one is bigger than the other?

I don't know where 'infinity and jelly donuts' comes from, I'm afraid.
6. 22 May '05 04:23
'Infinity and Jelly Doughnuts' was a TV show i think that was aired ages ago.
7. 22 May '05 14:25
Originally posted by phgao
'Infinity and Jelly Doughnuts' was a TV show i think that was aired ages ago.
I thought it was the name of an episode of a tv show aired ages ago.
8. 22 May '05 21:30
Originally posted by davegage
Which of the following sets contains the largest number of elements? Which of the following sets contains the smallest number of elements?
Rank the sets in order of number of elements from smallest to largest.

0. The set of all positive whole numbers and 0: {0,1,2,...}
1. The set of all positive whole numbers: {1,2,3,...}
2. The set of all positive m ...[text shortened]... ing it up on the internet, where does the expression 'Infinity and Jelly Doughnuts' come from?
9. 26 May '05 01:12
Originally posted by mantawa
Quite possibly.
10. 26 May '05 02:02
Originally posted by davegage
Which of the following sets contains the largest number of elements? Which of the following sets contains the smallest number of elements?
Rank the sets in order of number of elements from smallest to largest.

0. The set of all positive whole numbers and 0: {0,1,2,...}
1. The set of all positive whole numbers: {1,2,3,...}
2. The set of all positive m ...[text shortened]... ing it up on the internet, where does the expression 'Infinity and Jelly Doughnuts' come from?
OK, lemme see. Here's what I think I know:

1. Sets 0, 1 and 2 are all countably infinite, so they're the same "size".
2. Sets 3, 4 and 5 are uncountably infinite (I'm assuming you're talking about real numbers), which is "bigger" than countably infinite.

Here's what I might know:

3. Set 6 is uncountably infinite, and I believe it's the same "size" as sets 3-5.

Here's what I definitely don't know:

4. I think 7 is also uncountably infinite, but I'm not sure if it's bigger than sets 3-5 or not. I think it is, but I'm sure I don't know...

So I think set 7 is the "biggest", unless it's the same "size" as sets 3-5 (and I think set 6).

Oh, and I looked up the Infinity and Jelly Doughnuts thing on the internet. ("Who said that?!? Who said che@ter?!?"😉. It's from Magnum PI, the kickinest Hawaiian detective show since Five-0. Which I never saw. Hawaii Five-0, not Magnum PI. But I assume it too was kickin'.
😕
11. 26 May '05 02:08
Originally posted by Acolyte
Possibly, except that summing {-infinity,1,2,3,...} is impossible in any normal sense.

Suppose all I care about is the order of the elements within a set, so {1,2,3} is the same as {4,10,357}. How does this make the two sets I gave before different, and it what way can I say one is bigger than the other?

I don't know where 'infinity and jelly donuts' comes from, I'm afraid.
Does {-infinity,1,2,3,...} mean the set of all integers? That's what I'm assuming. I'm not sure if I understand the question, but if you compare each element of the sets element by element, I suppose the positive integer set is "bigger" because each element is further right on the number line. That's what I see when I look at your other example sets {1,2,3} and {4,10,357} anyway.

But I'm not sure if you can compare them that way as they are, because they're no obvious starting point --> (-infinity) vs. 1? It doesn't make much sense.

Is that what you meant?
12. 26 May '05 08:17
Originally posted by davegage

For extra credit: Without looking it up on the internet, where does the expression 'Infinity and Jelly Doughnuts' come from?
I'm pretty sure it's from our local small town police , who have always and will always just sit on their arses eating doughnuts all night .