CMSC250, Spring 2004 Homework 12

Due Wednesday, April 28 at the beginning of your discussion section.

You must write the solutions to the problems single-sided on your own lined paper, with all sheets stapled together, and with all answers written in sequential order or you will lose points.

1. Prove or disprove the following:
   1. $f : (0, \infty) \to {\bf {R}}$ defined by $f(x) = \ln(x)$ is 1-1.
   2. $f : (0, \infty) \to {\bf {R}}$ defined by $f(x) = \ln(x)$ is onto.

2. Let $A$ and $B$ be sets, and $f : A \to B$ be the function $f(x) = x^2$. For each of the four following questions, define the sets $A$ and $B$ such that $f$ is
   1. 1-1 and onto.
   2. 1-1 but not onto.
   3. onto but not 1-1.
   4. neither 1-1 nor onto.
   Note: You do not need to prove that each of your definitions of $A$ and $B$ has the desired properties; just give the sets.

3. Let $f$ be an arbitrary bijective function from ${\bf {R}}$ to ${\bf {R}}$.
   1. Define a function $g_1 : {\bf {R}}\to {\bf {R}}$ in terms of $f$ so that $g_1$ is 1-1 but not onto. Prove that $g_1$ satisfies this criteria.
   2. Define a function $g_2 : {\bf {R}}\to {\bf {R}}$ in terms of $f$ so that $g_2$ is onto but not 1-1. Prove that $g_2$ satisfies this criteria.

4. Let $D$ denote the set of odd integers.
   That is, $D = \{n \in {\bf {Z}}\mid n=2k+1$    for some integer $k\}$.
   1. Define a bijective function $h_1 : D \to {\bf {Z}}$. Prove that $h_1$ is a bijection.
   2. Define a bijective function $h_2 : {\bf {Z}}\to D$. Prove that $h_2$ is a bijection.

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