3 ms·
I'm interested. What's the solution?
by rkaplan 13y ago
I'm interested. What's the solution?
- daturkel 13y agoHere's a link to the proof I typed up for my problem set: https://www.sharelatex.com/project/534230c0234f079f3ce526fa?r=035466af&rs=ps&rm=d https://www.sharelatex.com/project/534230c0234f079f3ce526fa?... Theorem 1.10 states "If S andT are countable sets, the set S ×T ={(s,t) : s∈S,t∈T} is countable." and is proven using a diagonal argument.
- nmrm 13y agoMakes perfect sense. I'll code something up next weekend.
- cousin_it 13y agoThe set of all possible submarine routes is countable (velocity vector + position at time 0). Let's assume we have an enumeration of that set. Now it's easy to bomb them all. At time 1 we bomb the point where submarine 1 is at time 1, at time 2 we bomb the point where submarine 2 is at time 2, and so on.
- daturkel 13y agoYep, this is how the proof works!
- Someone 13y agoHow does that use the diagonal argument? If the diagonal argument could be used it would be to prove that the number of submarine routes isn't countable. Edit: just looked at the SVG called "Diagonal Argument.svg". That's not what I know as the diagonal argument (https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument, http://mathworld.wolfram.com/CantorDiagonalMethod.html http://mathworld.wolfram.com/CantorDiagonalMethod.html)