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  3. 2 tk ex 87 use the following deep theorem of...

Question: 2 tk ex 87 use the following deep theorem of...

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2. [TK, Ex 8.7] Use the following deep theorem of Banach and Tarski to show that the four conditions below cannot all hold. Theorem 1. The unit ball in R3 can be decomposed into a finite number of pieces which may be reassembled, using only translation and rotation, to form two disjoint copies of the unit ball. (a) Every bounded set, E, in R3 has a volume, μ(E), with μ(E) 2 0. (b) Let E be a bounded set in R3 and let F be a rotation and translation of E. Then (c) If E is a cube of side length a then (E)-a (d) If Ei and E2 are disjoint bounded sets in R3, then μ(El U E2)-μ(E) + μ(6).

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