CMSC 250 Homework 10 Fall 2001
Due Wed Nov 7 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. Let $X$ and $Y$ be nonempty finite sets, and let $f$ be a function with domain $X$ and codomain $Y$.

    1. If $f$ is onto, what can you say about the sizes of $X$ and $Y$?

    2. If $f$ is one-to-one, what can you say about the sizes $X$ and $Y$?

    3. If $f$ is one-to-one and onto, and $n(X)=(n(Y))^{2}$, what is $n(X)$?

    4. What must be true about $X$ and $Y$ if $f$ has an inverse function from $Y$ to $X$?

  2. Prove that


    \begin{displaymath}\frac{log_{2}3}{log_{2}4}=\frac{log_{5}3}{log_{5}4}.\end{displaymath}

  3. Social Security numbers consist of 9 digits. What is the maximum number of people who can be assigned unique Social Security numbers assuming that there are no other restrictions on Social Security numbers?

  4. I have a bag filled with marbles. There are 17 red marbles, 12 blue marbles, 14 green marbles, 9 yellow marbles, and 3 white marbles. How many marbles would I have to pick out of the bag to guarantee that I have 5 marbles of the same color?

  5. Find the inverse of $f(x)=x^{3}+ 2$, $x\in{\bf {R}}$.

  6. Let $D = \{x\in{\bf {Z}}\vert 0\leq x < 12\}$. Find examples of functions for each of the parts below. Find algebraic expressions for the functions. Do not list ordered pairs of elements.

    1. Find $f:D\rightarrow D$ such that $f(f(x)) = 0$, but $f(y)\neq 0$ for some $y\in D$.

    2. Find $g:D\rightarrow D$ such that $g(g(g(g(x)))) = x$.

    3. Find $h:D\rightarrow D$ such that $h(x)=h^{-1}(x)$.

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Deep Saraf
2001-10-31