This exercise is designed to give some insight into the Banach–Tarski paradox. Find each of the following without using the axiom of choice.
(a) A subset of IR that is congruent to a proper subset of itself.
(b) A bounded subset of the plane that is congruent to a proper subset of itself.
(c) A bounded subset S of the plane such that S is the disjoint union of two nonempty sets S1 and S2, and where S1 and S2 are both congruent to S.
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