MAT511 homework,         due Nov 12, 2003

1. Suppose that $A$ is a finite set with $m$ elements, and $B$ is a finite set with $n$ elements.
   1. Find the total number of functions from $A$ to $B$ if
      • $m=n$
      • $m>n$
      • $m<n$
   2. Find the number of one-to-one functions from $A$ to $B$ if
      • $m=n$
      • $m>n$
      • $m<n$

2. Give an example of functions $f:A\rightarrow B$ and $g:B\rightarrow C$, (be sure to specify domains and ranges) for which
   1. $g$ is onto $C$, but $g\circ f$ is not onto $C$.
   2. $g\circ f$ is onto $C$, but $f$ is not onto $B$.
   3. $g$ is one-to-one, but $g\circ f$ is not one-to-one.
   4. $g\circ f$ is one-to-one but $g$ is not one-to-one.

3. Let $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$ be given by $f(x) = x^2 + 1$. Find the following (remember that in this context $[a,b]$ is the set $\left\{{x\in{\mathbb{R}} \vrule{} a \le x \le b}\right\}$).
   1. $f( [1,2] )$
   2. $f( [-1,2] )$
   3. $f^{-1}( [5,10] )$
   4. $f^{-1}( [-1,5] \cup [17,26] )$

4. Let $f:A\rightarrow B$, and $D\subseteq A$, $E\subseteq B$. Prove that $D \subseteq f\left( f^{-1}(D) \right)$. Also, Give an example where $D \ne f\left( f^{-1}(D) \right)$.