14 Aug 06
How about...
A confused bank teller transposed the dollars and cents when he cashed a check for Ms Smith, giving her dollars instead of cents and cents instead of dollars. After buying a newspaper for 50 cents, Ms Smith noticed that she had left exactly three times as much as the original check. What was the amount of the check? (Note: 1 dollar = 100 cents.)
Purely mathematical!
15 Aug 06
Originally posted by JamdogThx!
How about...
A confused bank teller transposed the dollars and cents when he cashed a check for Ms Smith, giving her dollars instead of cents and cents instead of dollars. After buying a newspaper for 50 cents, Ms Smith noticed that she had left exactly three times as much as the original check. What was the amount of the check? (Note: 1 dollar = 100 cents.)
Purely mathematical!
15 Aug 06
7 edits
Originally posted by chess kid1It ain't easy, kid. And why didn't you thank me for mine? :'( But just to prove that I'm not really upset, here's another one for your scrapbook:
Thx!
There is a subway line from the airport to the Hilbert hotel which operates as follows: there is a station at each ordinal number, and every station is assigned a unique ordinal. The subway stops at each station, in order. At each station people disembark and board, in order, as follows:
i) if any passengers are on the subway, exactly 1 disembarks, then
ii) aleph_0 passengers board the subway.
Station 0 is at the airport, and the Hilbert hotel is at station w_1 (the first uncountable cardinal). The subway starts its journey empty. Aleph_0 passengers board the subway to the Hilbert hotel at the airport (station 0), and off it goes.
When the subway pulls up to the Hilbert hotel at station w_1, how many passengers are on it? Is it 0, aleph_1, some determinate value in between, or indeterminate?
15 Aug 06
Originally posted by ThudanBlunderyour conditions don't make sense. there are aleph_1 stations but each has a unique cardinal number??
It ain't easy, kid. And why didn't you thank me for mine? :'( But just to prove that I'm not really upset, here's another one for your scrapbook:
There is a subway line from the airport to the Hilbert hotel which operates as follows: there is a station at each ordinal number, and every station is assigned a unique ordinal. The subway stops at ea ...[text shortened]... ssengers are on it? Is it 0, aleph_1, some determinate value in between, or indeterminate?
17 Aug 06
Originally posted by ThudanBlunderI'm not sure if this is correct/rigorous enough, as set theory isn't really my field, but as a few first tentative steps...
It ain't easy, kid. And why didn't you thank me for mine? :'( But just to prove that I'm not really upset, here's another one for your scrapbook:
There is a subway line from the airport to the Hilbert hotel which operates as follows: there is a station at each ordinal number, and every station is assigned a unique ordinal. The subway stops at ea ...[text shortened]... ssengers are on it? Is it 0, aleph_1, some determinate value in between, or indeterminate?
The ordinals are arranged in increasing order, and we can think of them each as subsets of each of their successors, and supersets of all their predecessors. ie.
1 subset of 2 subset of 3... subset of w...subset of w + 1...subset of w + 2...subset of w.2....subset of 3.w + 5... etc.
So there are only countably many ordinals before the first uncountable ordinal.
Since a countable number of countable sets is countable, I would say there is a countable number of people on the train.
I would say this is determinate, unless there's some axiom of choice trickery going on that I've missed.
17 Aug 06
1 edit
Originally posted by aginisOriginally posted by aginis
it is exactly what you wrote
your conditions don't make sense. there are aleph_1 stations but each has a unique cardinal number??
It? Do you mean the above? Hardly exact.
'Ordinal' denotes position whereas 'cardinal number' denotes size.
Bearing in mind that we are talking about transfinite ordinals, what exactly is your objection?
17 Aug 06
1 edit
Originally posted by ThudanBlundersorry i mixed up the word cardinal and ordinal
Originally posted by aginis
[b]your conditions don't make sense. there are aleph_1 stations but each has a unique cardinal number??
It? Do you mean the above? Hardly exact.
'Ordinal' denotes position whereas 'cardinal number' denotes size.
Bearing in mind that we are talking about transfinite ordinals, what exactly is your objection?[/b]
my problem is that you imply that there exists a one to one function from the set of stations (S) to the natural numbers. At the same time you are stating that the set S has cardinality aleph_1. thus a one to one function exists from a set of cardinality aleph_1 to a set of cardinality aleph_0. This is impossible.
In other words if each station has a unique ordinal number attached to it then the set of stations must be countable (for example in order from least to greatest)
20 Aug 06
Originally posted by aginisI thought that is what you meant.
sorry i mixed up the word cardinal and ordinal
my problem is that you imply that there exists a one to one function from the set of stations (S) to the natural numbers. At the same time you are stating that the set S has cardinality aleph_1. thus a one to one function exists from a set of cardinality aleph_1 to a set of cardinality aleph_0. This is impossibl ...[text shortened]... d to it then the set of stations must be countable (for example in order from least to greatest)
However, here we are not dealing with natural numbers but with transfinite numbers.
http://mathworld.wolfram.com/TransfiniteNumber.html
27 Aug 06
1 edit
Originally posted by XanthosNZ
You attend a party along with N other people. Given any group of 4 party-goers you can be sure that at least one of the 4 knows the other three. Prove that at least one person knows all others at the party.
EDIT: I have saved this thread.
27 Aug 06
Originally posted by tomtom232Originally posted by aginis
assume that no guest knows all the other guests. then each guest does not know (at least) one other person. since in any group of 4 people you must know at least 1 there are a maximum of 2 strangers per guest. now according to our assumption if i pick a random guest A from the party there Exists guest B such that A,B are strangers. now we pick a third guest C if A,B,C are all strangers then they've reached their maximum of two strangers each so any fourth guest D must know them all (If there are only 4 guests then we are done) but again according to our assumption there Exists a fifth guest E who is a stranger to D. take the group A,B,D,E. A is a stranger to B who is a stranger to A and D is a stranger to E who is a stranger to D, this contradicts the assumption so at least one guest knows everyone.
in fact this shows that if A,B,C are all strangers then all the other guests know everyone at the party.