Why are the cardinality of $\mathbb{R^n}$ and $\mathbb{R}$ the same? As I study the first part of abstract algebra, I have a question: why $\left\vert{\mathbb R}\right\vert = \left\vert{\mathbb R^2}\right\vert$?  And moreover, I knew the fact that $\left\vert{\mathbb R}\right\vert = \left\vert{\mathbb R^n}\right\vert$.  Does the equality apply to any infinite set? How can I prove them?
Cardinality of the Cartesian Product of Two Equinumerous Infinite Sets Is the cardinality of the Cartesian product of two equinumerous infinite sets the same as the cardinality of any one of the sets? I couldn't find this explicitly stated in any handout or text. This certainly seems to be true from the examples I have seen:  The Cartesian product of two infinitely countable sets is again infinitely countable. The Cartesian product of two sets with cardinality of continuum again has cardinality of continuum.
Is a semigroup with unique right identity and left inverse a group? We know that a semigroup with a right identity and right inverse for all elements is a group (e.g. see ). Symmetrically, also a left identity together with a left inverse implies a group. We also know that a semigroup with a right identity and a left inverse is NOT necessarily a group (see ). My questions are:   in a semigroup, is the existence of a UNIQUE right identity together with the existence of a left inverse enough to have a group? in a semigroup, is the existence of a right identity together with the existence of a UNIQUE left inverse enough to have a group?   I think both these claims are false,  ut haven't found a counter-example so far.