Exercises

1. Let $f : A\rightarrow B$ and let $\{E_\alpha\}_{\alpha\in \Lambda}$ be a collection of subsets of $A$. Prove that
   1. $f[\cup_\alpha E_\alpha]=\cup_\alpha f[E_\alpha]$,
   2. $f[\cap_\alpha E_\alpha]\subset \cap_\alpha f[E_\alpha]$.

2. Let $f : A\rightarrow B$ and let $\{F_\alpha\}_{\alpha\in \Lambda}$ be a collection of subsets of $B$. Prove that
   1. $f^{-1}[\cup_\alpha F_\alpha]=\cup_\alpha f^{-1}[F_\alpha]$,
   2. $f^{-1}[\cap_\alpha F_\alpha]=\cap_\alpha f^{-1}[F_\alpha]$.

3. Construct examples of the following:
   1. A mapping which is not one-to-one and not onto.
   2. A mapping which is not one-to-one, but is onto.
   3. A mapping which is one-to-one, but is not onto.

4. Construct an example of a mapping $f : A\rightarrow B$ such that $f[E \cap F] \not= f[E] \cap f[F]$, where $E,F\subset A$.

5. Using the notation in problem 1, show that if $f$ is one-to-one, then $f[\cap_\alpha E_\alpha] = \cap_\alpha f[E_\alpha]$.

6. Let $f : A\rightarrow B$. Show that

   (a) if $f$ is one-to-one then $f^{-1}[f[A]] = A$,

   (b) if $f$ is onto, $f(f^{-1}(B)) = B$.

7. Let $A \stackrel{f}{\rightarrow} B\stackrel{g}{\rightarrow} C$. Show that if $f$ is one-to-one and onto and if $g$ is one-to-one and onto, then $gf$ is one-to-one and onto.

8. Let $A \stackrel{f}{\rightarrow} B\stackrel{g}{\rightarrow} C \stackrel{h}{\rightarrow} D$. Show that $(hg)f = h(gf)$. (Composition is associative).

9. Give examples of relations on a set $S$ which satisfy all but one of the axioms for an equivalence relation on $S$.

10. Determine the equivalence classes for Example 1.1.3.

11. Show that if $S\not= \emptyset$ has a partition into disjoint nonempty subsets, then an equivalence relation may be defined on $S$ (actually find this equivalence relation and show that it is an equivalence relation) for which the subsets of the partition are the equivalence classes. (Converse of Theorem 1.1.4)

12. Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be the map given by $f(x) = x^2$. Let
    
    $$A=[1,2]=\{x\in \mathbb{R}\ \vert\ 1\leq x\leq 2\},$$
    
    
    
    
    $$B=(-1,1)=\{x\in \mathbb{R}\ \vert\ -1< x <1\},$$
    
    
    
    
    $$C=(4,9)=\{x\in \mathbb{R}\ \vert\ 4< x <9\},$$
    
    
    
    
    $$D=[0,9]=\{x\in \mathbb{R}\ \vert\ 0\leq x \leq 9\}.$$
    
    
    Find

    (a) $f[A]$,

    (b) $f[B]$,

    (c) $f^{-1}[C]$,

    (d) $f^{-1}[D]$,

    (e) a nonempty set $E \subset \mathbb{R}$ such that $f^{-1}[E] = \emptyset$.