CMSC 250

Quiz #9 KEY

Monday, Nov. 5, 2001

Write all answers legibly in the space provided. The number of points possible for each question is indicated in square brackets - the total number of points on the quiz is 30, and you will have exactly 15 minutes to complete this quiz. You may not use calculators, textbooks or any other aids during this quiz.

[18 pnts.] Answer each of the following by writing either the word ``Yes'' or the word ``NO'' into the blank provided. Determine if the following functions are one-to-one. Grading: 3 points each - all or nothing
1. YES $f:Z \rightarrow Z$ defined as $\forall n \in Z, f(n)= n - 1$
2. NO $g:R \rightarrow Z$ defined as $\forall n \in R, g(n)= \lceil n \rceil$
3. YES $h:D \rightarrow C$ where D = $\{1,3,5,7\}$ and C = $\{2,4,6, 8\}$
  defined as $h = \{(1,4), (3,6), (5,2), (7,8)\}$
Determine if the following functions are onto.
1. YES $f:Z \rightarrow Z$ defined as $\forall n \in Z, f(n)= n - 1$
2. YES $g:R \rightarrow Z$ defined as $\forall n \in R, g(n)= \lceil n \rceil$
3. YES $h:D \rightarrow C$ where D = $\{1,3,5,7\}$ and C = $\{2,4,6, 8\}$
  defined as $h = \{(1,4), (3,6), (5,2), (7,8)\}$
[12 pnts.] For each of the following determine if it a bijection (one-to-one and onto). If it is, determine the value of the inverse function from . If it is not, give the reason it is not a bijection. Grading: 4 points each - -4 if the yes/no is wrong (interpret if unstated) and -2 if the inverse function or reason is wrong or missing
1. YES
  
  $x = \frac{4-y}{3}$
  
  $f^{-1}(y) = \frac{4-y}{3}$
2. NO
  because the negative elements of the codomain are not reached.
3. YES
  
  $\sqrt[3]{y} = x$
  
  $f^{-1}(y) = \sqrt[3]{y}$