one-to-one correspondence
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one-to-one correspondence,<a href="http://www.lightbulbschina.com">Energy <a href="javascript:;" onClick="javascript:tagshow(event, 'saving');" target="_self"><u><strong>saving</strong></u></a> lamp</a>one-to-one correspondenceThe new manager, however, has everything#x under control. Instead of tending to just oneGeneral Purpose Bladebus, he zigs and zags through the diagram, fanning out from the corner as shown below.He starts with passenger 1 on bus 1 and gives her the first empty room. The second and third empty rooms go to passenger 2 on bus 1, followed by passenger 1 on bus 2, both of whom are depicted on the second diagonal from the corner of the diagram. After serving them, the manager proceeds to the third diagonal and hands out a set of room keys to passenger 1 on bus 3, passenger 2 on bus 2, and passenger 3 on bus 1.I hope the manager’s procedure ― progressing from one diagonal to another ― is clear from the picture above, and that you’re convinced that any particular person will be reached in a finite number ofGeneral Purpose Bladesteps.So, as advertised, there’s always room at the Hilbert Hotel.The argument I’ve just presented is a famous one in the theory of infinite sets. Cantor used it to prove that there are exactly as many positive fractions (ratios p/q of positive whole numbers p and q) as there are natural numbers (1, 2, 3, 4, …). That’s a much stronger statement than saying both sets are infinite. It says they are infinite to precisely the same extent, in the sense that a “one-to-one correspondence” can be established between them.You could think of this correspondence as a buddy system in which each natural number is paired with some positive fraction, and vice versa. The existence of such a buddy system seems utterly at odds with common sense ― it’s the sort of sophistry that made Poincaré recoil. For it implies we could make an exhaustive list of all positive fractions, even though there’s no smallest one!And yet there is such a list. We’ve already found it. The fraction p/q corresponds to passenger p on bus q, and the argument above shows that each of these fractions can be paired off #xwith a certainGeneral Purpose Bladenatural number 1, 2, 3,…, given by the passenger’s room number at the Hilbert Hotel.相关的主题文章：
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