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

Question: Let $\mathit{f}\mathbf{:}\mathit{R}\mathbf{\to }\mathit{R}$and $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{}\mathbf{>}\mathbf{}\mathbf{0}$let for all $\mathit{x}\mathbf{\in }\mathit{R}$. Show that f(x) $\mathbf{}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$is strictly increasing if and only if the function$\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{/}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ is strictly decreasing.

Prove that \({\bf{P}}\left( {\bf{A}} \right) \subseteq {\bf{P}}\left( {\bf{B}} \right)\) if and only if \({\bf{A}} \subseteq {\bf{B}}\).

Determine whether each of the strings of 12 digits is a valid UPC code.

a) 036000291452

b) 012345678903

c) 782421843014

d) 726412175425

Let\(A = \left\{ {0,2,4,6,8,10} \right\}\),\(B = \left\{ {0,1,2,3,4,5,6} \right\}\)and\(C = \left\{ {4,5,6,7,8,9,10} \right\}\).Find

a)\(A \cap B \cap C\)

b)\(A \cup B \cup C\)

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

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

Let$f:R\to R$ and let$f\left(x\right)>0$ for all $x\in R$. Show that f(x) is strictly increasing if and only if the functionrole="math" localid="1668414567143" $g\left(x\right)=1/f\left(x\right)$ is strictly decreasing.

Prove that if it is possible to label each element of an infinite set S with a finite string of keyboard characters, from a finite list characters, where no two elements of S have the same level, then S is a countably infinite set.

To Determine a formula for the probability of \({E_1} \cup {E_2} \cup {E_3}\).

Prove that if n is an odd integer, then$⌈n^2/4⌉=\frac{\left(n^2+3\right)}{4}$

For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence.

$\begin{array}{r}3,6,11,18,27,38,51,66,83,102,\dots \\ 7,11,15,19,23,27,31,35,39,43,\dots \\ 1,10,11,100,101,110,111,1000,1001,1010,1011,\dots \\ 1,2,2,2,3,3,3,3,3,5,5,5,5,5,5,5,\dots \\ 0,2,8,26,80,242,728,2186,6560,19682,\dots \mid \\ 1,3,15,105,945,10395,135135,2027025,34459425,\dots \\ 1,0,0,1,1,1,0,0,0,0,1,1,1,1,1,\dots \\ 2,4,16,256,65536,4294967296,\dots \end{array}$

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

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

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

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