Past exam of the logic and set theory course of the University of Oxford / 2007 / B / 7 / ii / Solution

The set of all functions from a set of size $\mathfrak{c}$ to another set of size $\mathfrak{c}$ has size ${\mathfrak{c}}^{\mathfrak{c}}$. Cardinal exponentiation and the law $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \mu }$ giveThereforeThe lower bound can also be seen by taking characteristic functions of all subsets of $\mathbb{C}$. Cantor's theorem makes this strictly larger than $\mathfrak{c}$, unlike the continuous-function space in the next part.