## Probability

### The Analysis of Data, volume 1

Set Theory: Exercises

## A.6. Exercises

1. Prove the assertions in Example A.1.1.
2. Prove the assertion in Example A.1.6.
3. Prove that $f(U\cap V) = f(U)\cap f(V)$ is not true in general.
4. Let $A_0=\{a\}$ and define $A_k=2^{A_{k-1}}$ for $k\in\mathbb{N}$. Write down the elements of the sets $A_k$ for all $k=1,2,3$.
5. Let $A,B,C$ be three finite sets. Describe intuitively the sets $A^{(B^C)}$ and $(A^B)^C$. What are the sizes of these two sets?
6. Let $A$ be a finite set and $B$ be a countably infinite set. Are the sets $A^{\infty}$ and $B^{\infty}$ countably infinite or uncountably infinite?
7. Find a sequence of sets $A_n$, $n\in\mathbb{N}$ for which $\liminf A_n\neq \limsup A_n$.
8. Describe an equivalence relation with an uncountably infinite set of equivalence classes, each of which is a set of size 2.