CMSC 250 Homework 13 Spring 2006

Due Wed May 3 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 $X = \{1,3,5,7\}$ and $Y = \{s,t,u,v,w\}.$ Define $f: X \rightarrow Y$ by the following arrow diagram

   

   1. write the domain and the co-domain of f.

   2. what is the range of $f$?

   3. what is the inverse image of s? Of u?

2. Define f: $R \rightarrow R$ by the rule

   $f(x) = 3x + 5$ for all $x \in R$.

   1. Is $f$ onto? Prove or give a counterexample.

   2. Is $f$ one-to-one? Prove or give a counterexample.

   3. Find its inverse function or state why no such function can exist.

3. 1. How many one-to-one functions are there from a set with three elements to a set with four elements?

   2. How many one-to-one functions are there from a set with six elements to a set with five elements?

   3. How many onto functions are there from a set with three elements to a set with two elements?

   4. How many onto functions are there from a set with four elements to a set with two elements?

4. 1. In a group of 687 people, must there be two who have same first name and last name initials? Why or why not?

   2. If 5 distinct integers are chosen from between 2 and 10 inclusive, must at least 1 of them be a prime? Why or why not?

   3. I have a bag filled with marbles. There are 15 red marbles, 9 blue marbles, 12 green marbles, 15 yellow marbles, 4 black 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. If each function F and G are defined by formulas find $G\circ F$ and $F \circ G$ and determine whether $G\circ F$ equals $F \circ G$
   1. F(x)$= x^2$; G(x)$= x^2+1$

   2. F(x)$= x^3$; G(x)$= x^3$.

6. Let $D = \{x\in{\bf {Z}}\vert 0\leq x < 16\}$. Find examples of functions for each of the parts below. Find algebraic expressions for the functions. You may wish to use algebraic equations taken mod $16$ to construct your functions. Do not list ordered pairs of elements, or the identity function for any of your answers.
   1. Find $h:D\rightarrow D$ such that $h(h(h(x)))=h^{-1}(x)$.

   2. Find $f:D\rightarrow D$ such that $\forall x\in D,\,\,f(x)=f(x+5)$, and $f$ is not a constant function.

   3. Find $g:D\rightarrow D$ such that $f(f(x))=0$, but $\exists x\in D,\,\, f(x)\neq 0$.

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