Banach-Tarski paradox

In the present work we study the Banach-Tarski Paradox and other paradoxical decompositions of sets, groups and semigroups. These decompositions are described especially using free groups and semigroups. We can construct such groups using words made of letters. We study both finite and denumerable paradoxical decomposition. Further we deal with equidecomposability, which we need for a proof of Banach-Tarski Paradox. We present a proof of Banach-Schröder-Berstein Theorem