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  that is congruent to a proper subset of...

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
 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.