Cartesian product power set and functions

Statement

Let $A = \{0, 1, 2\}$. Find the cardinality of: (i) the cartesian product $A^2 = A \times A$; (ii) the power set $\mathcal{P}(A)$; (iii) the set $A^A$ of functions from $A$ to $A$.

Solution

$A$ has $3$ elements. (i) $A^2$ consists of ordered pairs $(x,y)$ with $x, y \in A$: $3 \cdot 3 = \boxed{9}$ pairs. (ii) $\mathcal{P}(A)$ has $2^{3} = \boxed{8}$ subsets. (iii) A function $A \to A$ assigns to each of the $3$ elements one of $3$ images: $3^{3} = \boxed{27}$ functions.