- #1

ahrkron

Staff Emeritus

Science Advisor

Gold Member

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Consider set B and let U be the subset consisting of all points that are a positive integer number of radians clockwise from X along the circle. This is a countably infinite set (the irrationality of Pi prevents two such points from coinciding). Let set V be everything else.

If you pick set U up and rotate it counterclockwise by one radian, something very interesting happens. The deleted hole at X gets filled by the point 1 radian away, and the point at the (n-1)-th radian gets filled by the point at the n-th radian. Every point vacated gets filled, and in addition, the empty point at X gets filled too!

Thus, B may be decomposed into sets U and V, which after this reassembling, form set A, a complete circle!

I just couldn't resist posting this.