Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

Question: a) Prove that a strictly increasing function from R to itself is one-to-one.

b) Give an example of an increasing function from R to itself is not one-to-one.

Show that if \({\bf{A}} \subseteq {\bf{C}}\)and \({\bf{B}} \subseteq {\bf{D}}\), then\({\bf{A \times B}} \subseteq {\bf{C \times D}}\).

$\mathit{A}\mathbf{=}\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]\mathbf{\text{and}}\mathit{B}\mathbf{=}\left[\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\end{array}\right]$Let Find

a) $\mathbf{A\nu B}$.

b) $\mathbf{A\Lambda B}$.

c) $\mathit{A}\mathbf{\odot }\mathit{B}$.

Draw the Venn diagrams for each of these combinations of the sets A, B, and C.

a) \(A \cap \left( {B \cup C} \right)\)

(b)\(\overline A \cap \left( {\overline B \cap \overline C } \right)\)

(c)\(\left( {A - B} \right) \cup \left( {A - C} \right) \cup \left( {B - C} \right)\)

Use Exercise 25 to providea proof different from that in the text that the set of rational numbers is countable. [Hint: Show that you can express a rational number as a string of digits with a slash and possibly a minus sign].

Prove that if m and n are positive integers and x is a real number, then

$⌊\frac{\lfloor x\rfloor +n}{m}⌋=\left[\frac{x+n}{m}\right.⌋$

Show that if denotes the positive integer that is not a perfect square, then$\mathrm{an}=\mathrm{n}+\left\{\mathrm{n}\right\}$ where {x} denotes the integer closest to the real number x

Let $\mathit{A}\mathbf{=}\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]\mathbf{\text{and}}\mathit{B}\mathbf{=}\left[\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]$

Find

a) $\mathit{A}\mathbf{\vee }\mathit{B}$.

b) $\mathit{A}\mathbf{\wedge }\mathit{B}$.

c)$\mathit{A}\mathbf{\odot }\mathit{B}$.

Draw the Venn diagrams for each of these combinations of sets A, B, and C.

a) \(A \cap \left( {B - C} \right)\)

b) \(\left( {A \cap B} \right) \cup \left( {A \cap C} \right)\)

c)\(\left( {A \cap \mathop B\limits^\_ } \right) \cup \left( {A \cap \mathop C\limits^\_ } \right)\)

Determine which transposition errors the check digit of a UPC code finds. Some airline tickets have a 15 -digit identification number \({a_1}{a_2} \ldots {a_{15}}where{a_{15}}\) is a check digit that equals \({a_1}{a_2} \ldots {a_{14}}\,mod\,7.\)