Past exam of the logic and set theory course of the University of Oxford / 2009 / B / 5 / iii / Solution

The notation ${}^{\mathbb{R}}\mathbb{R}$ means all functions from the real numbers to themselves, with no continuity restriction. Its cardinal isThe exponent law follows by viewing a function to a function set as one function on a Cartesian product. Cantor's theorem gives $2^{\mathfrak{c}}>\mathfrak{c}$, so this is strictly larger than the other continuum-sized examples here. Characteristic functions of subsets of $\mathbb{R}$ already supply the lower bound $2^{\mathfrak{c}}$.