MAT511 homework,         due Nov 5, 2003

1. Give a relation $R$ from $A = \left\{{5, 6, 7}\right\}$ to $B=\left\{{3, 4, 5}\right\}$ such that
   1. $R$ is not a function.
   2. $R$ is a function from $A$ to $B$, with the image of $R$ equal to $B$.
   3. $R$ is a function from $A$ to $B$, with the image of $R$ not equal to $B$.
   4. $R$ is a function from $A$ to $B$ which is not one-to-one.

2. Explain why the functions
   $$f(x) = \frac{9-x^2}{x+3}$$   and$$\qquad g(x)=3-x$$
   are not equal.

3. A metric on a set $X$ is a function $d: X \times X \rightarrow {\mathbb{R}}$ so that for all $x$, $y$, and $z$ in $X$, the following properties are satisfied:
   • $d(x,y) \ge 0$
   • $d(x,y) = 0$ if and only if $x=y$.
   • $d(x,y) = d(y,x)$
   • $d(x,y) + d(y,z) \ge d(x,z)$

   Prove that each of the following is a metric for the indicated set.

   the Euclidean metric

   $X={\mathbb{R}}$, $d(x,y) = \sqrt{(x-y)^2 }$

   the Manhattan metric

   $X={\mathbb{R}}^2$, $d\left( \strut (x,y), (z,w) \right) = \vert x-z\vert + \vert y-w\vert$

   the discrete metric

   $X$ is any set, $d(x,y) = 0$ whenever $x=y$, and $d(x,y)=1$ if $x\ne y$.

4. For each of the following, decide whether they are one-to-one and whether they are onto. Prove your answers.
   1. $f:{\mathbb{N}}\rightarrow {\mathbb{N}}$, $f(x)= 2x+1$
   2. $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$, $f(x)= 2x+1$
   3. $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$, $f(x)= 2^x$
   4. $f:{\mathbb{R}}\times{\mathbb{R}}\rightarrow {\mathbb{R}}$, $f(x,y)= x-y$
   5. $f:(1,\infty) \rightarrow (1,\infty)$, ${\displaystyle f(x)= \frac{x}{x-1}}$

5. Prove that if a real-valued function $f$ is strictly increasing, then $f$ is one-to-one. Also, give an example of a real-valued function $g$ which is strictly increasing, but is not onto.