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\begin{document}
\begin{center}
\Large{Exercises for Module 3: Set Theory II and Metric Spaces I}
\end{center}

1. Show that for sets $A,B \subseteq X$ and $f: X \to Y$, $f(A \cap B) \subseteq f(A) \cap f(B)$.

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2. Let $f: X \to Y$.. Prove that $f(f^{-1}(B)) \subseteq B$ holds for any $B \subseteq Y$. Furthermore, show that $f(f^{-1}(B)) = B$ holds for any $B \subseteq Y$ if and only if $f$ is surjective.

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3. Prove that $f(\cup_{i \in I}A_i) = \cup_{i \in I}f(A_i)$, where $f:X \to Y$, $A_i \subseteq X \, \forall i \in I$. 

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4. Show that $\N$ and $\Z$ have the same cardinality. 

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5. Show that $|(0,1)| =|(1,\infty)|$.

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6. Show that the infinity norm $||x||_\infty$, $x \in \R^n$, is a norm.

    
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7. Let $(X,d)$ be any metric space, and define $\tilde d: X \times X \to \R$ by 
    \begin{equation*}
        \tilde d(x,y) = \frac{d(x,y)}{1+d(x,y)}, \quad x,y \in X .
    \end{equation*}
    Show that $\tilde d$ is a metric on $X$.

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8. Let $X$ be a set and define $d\colon X \times X \to \R$ by $d(x,x) = 0$ and $d(x,y)=1$ for $x\neq y \in X$. Prove that $d$ is a metric on $X$. What do open balls look like for different radii $r>0$? %What does an arbitrary open set look like?

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