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

Show that if A is an infinite set, then it contains a countably infinite subset.

a) Show that the system of simultaneous linear equations

$\begin{array}{c}{a}_{11}{x}_1+a_{12}{x}_2+\dots +a_{1n}{x}_n=b_1\\ a_{21}{x}_1+a_{22}{x}_2+\dots +a_{2n}{x}_n=b_2\\ \dots \\ a_{n1}{x}_1+a_{n2}{x}_2+\dots +a_{nn}{x}_n=b_{n}n\end{array}$

in the variables $x_1,x_2,.....,x_n$can be expressed as AX = B, where$\mathit{A}\mathbf{=}\left[a_{ij}\right]\mathbf{,}\mathit{X}$is an $n\times 1$matrix with $x_i$the entry in its ith row, and Bis an $n\times 1$matrix with $b_i$the entry in its ith row.

b) Show that if the matrix $A=\left[a_{ij}\right]$is invertible (as defined in the preamble to Exercise 18), then the solution of the system in part (a) can be found using the equation$\mathit{X}\mathbf{=}{\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathit{B}$.

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) 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.

Let $f:R\to R$ and let$f\left(x\right)>0$ for all $x\in R$. Show that f(x) is strictly decreasing if and only if the functionrole="math" localid="1668414965156" $g\left(x\right)=1/f\left(x\right)$ is strictly increasing.

Let A, B, and C be sets. Show that (A − B) − C = (A − C) − (B − C).

Determine whether each of these sets is the power set of a set, where a and b are distinct elements?

(a) \(\phi \)

(b) \(\left\{ {\phi ,\;\left\{ a \right\}} \right\}\)

(c) \(\left\{ {\phi ,\left\{ a \right\},\left\{ {\phi ,a} \right\}} \right\}\)

(d) \(\left\{ {\phi ,\left\{ a \right\},\left\{ b \right\}\left\{ {a,b} \right\}} \right\}\)

Show that there is no infinite set A such that$\left|\mathrm{A}\right|<\left|{\mathrm{Z}}^{+}\right|={\mathrm{N}}_0$

Determine the check digit for the UPCs that have these initial 11 digits.

a) 73232184434

b) 63623991346

c) 04587320720

d) 93764323341

Find the sum and product of each of these pairs of num your answers as a hexadecimal expansion

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.