You have an ink stamp that is so amazingly precise that, when inked and pressed down on the plane, it makes every circle whose radius is an irrational number (centered at the center of the stamp) black.
Is it possible to use the stamp three times and make every point in the plane black?
If it is possible, where would you center the three stamps?
(In reply to
alternation implies 1-1 by Cory Taylor)
Perhaps it would be better to say there's an infinite number of rationals between any two different irrationals (and vice versa). (Does that clear anything up? :-) This way, you don't think of alternating points?
Another problem with thinking of "alternating" points... is this notion of adjacency. No point is adjacent to another point (unless, perhaps, you're naming the same point).
Either way, the notion of one being larger than the other is generally represented by sets. One can construct the (infinite) set of the rationals and the (infinite) set of irrationals, and then demonstrate that one can provide a mapping function from one to the other (but not vice versa).
Since we can't get a corresponding rational for every irrational, the set of irrationals is larger.
Methinks the proof of this is beyond the scope of flooble. :-)
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Did any of that help?
re: alternation implies 1-1
