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

Question: Show that the function $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\left|x\right|$ from the set of real numbers to the set of non-negative real numbers is not invertible, but if the domain is restricted to the set of non- negative real numbers, the resulting function is invertible.

Show that the set of all finite bit strings is countable.

Show that each of these properties is an invariant that isomorphic simple graphs either both have or both do not have.

a) connectedness

b) the existence of a Hamilton circuit

c) the existence of a Euler circuit

d) having crossing number C

e) having isolated vertices

f) being bipartite

Determine the value of $\prod _{k=1}^{100} \frac{k+1}{k}$.(The notation used here for products is defined in the preamble to Exercise 43 in Section 2.4.)

Find A + B, where

a)

$\begin{array}{r}\mathbf{A}\mathbf{=}\left[\begin{array}{ccc}1& 0& 4\\ -1& 2& 2\\ 0& -2& -3\end{array}\right]\\ \mathbf{B}\mathbf{=}\left[\begin{array}{ccc}-1& 3& 5\\ 2& 2& -3\\ 2& -3& -0\end{array}\right]\end{array}$

b)

$\begin{array}{r}\mathbf{A}\mathbf{=}\left[\begin{array}{cccc}-1& 0& 5& 6\\ -4& -3& 5& -2\end{array}\right]\\ \mathbf{B}\mathbf{=}\left[\begin{array}{cccc}-3& 9& -3& 4\\ 0& -2& -1& 2\end{array}\right]\end{array}$

Determine whether\({\bf{f}}\)is a function from\({\bf{Z}}\)to\({\bf{R}}\)if

a)\({\bf{f}}\left( {\bf{n}} \right) = \pm {\bf{n}}.\)

b)\({\bf{f}}\left( {\bf{n}} \right) = \sqrt {{{\bf{n}}^{\bf{2}}} + {\bf{1}}} .\)

c)\({\bf{f}}\left( {\bf{n}} \right) = {\bf{1}}/\left( {{{\bf{n}}^{\bf{2}}} - {\bf{4}}} \right).\)

Suppose that \(A\) is the set of sophomores at your school and \(B\) is the set of students in discrete mathematics at your school. Express each of these sets in terms of \(A\) and \(B\).

(a). the set of sophomores taking discrete mathematics in your school.

(b) the set of sophomores at your school who are not taking discrete mathematics

(c) the set of students at your school who either are sophomores or are taking discrete mathematics.

(d) the set of students at your school who either are not sophomores or not taking discrete mathematics.

Use set builder notation to give a description of each of these sets.

a) \(\left\{ {0,\;3,\;6,\;9,\;12} \right\}\)

b) \(\left\{ { - 3,\; - 2,\; - 1,\;0,\;1,\;2,\;3} \right\}\)

c) \(\left\{ {m,\;n,\;o,\;p} \right\}\)

What is the termof the sequence$\left\{a_n\right\}$if ${\mathbf{a}}_{\mathbf{n}}$equals

$\begin{array}{r}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\mathbf{}{\mathbf{2}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{?}\\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{7}\\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{}\mathbf{1}\mathbf{+}\mathbf{(}\mathbf{-}\mathbf{1}{\mathbf{)}}^{\mathbf{n}}\mathbf{?}\\ \mathbf{\left(}\mathbf{d}\mathbf{\right)}\mathbf{}\mathbf{-}\mathbf{(}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{n}}\mathbf{?}\end{array}$

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.

a) the integers greater than 10

b) the odd negative integers

c) the integers with absolute value less than 1,000,000

d) the real numbers between 0 and

e) the set $\mathbf{A}\mathbf{\times }{\mathbf{Z}}^{\mathbf{+}}$, where A = {2,3}

f ) the integers that are multiples of 10