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

Find the Boolean product of and, where $\mathit{A}\mathbf{=}\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 1\\ 1& 1& 1& 1\end{array}\right]$ and$\mathit{B}\mathbf{=}\left[\begin{array}{cc}1& 0\\ 0& 1\\ 1& 1\\ 1& 0\end{array}\right]$

What is the Cartesian product A X B, where A is the set of courses offered by the mathematics department at a university and B is the set of mathematics professors at this university? Give an example of how this Cartesian product can be used.

Show that the set $Z^{+}\times Z^{+}$is countable.

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

We define the Ulam numbers by setting $u_1=1$and $u_2=2$. Furthermore, after determining whether the integers less than n are Ulam numbers, we set n equal to the next Ulam number if it can be written uniquely as the sum of two different Ulam numbers. Note that $u_3=3,u_4=4,u_5=6,$and$u_6=8$

a) Find the first 20 Ulam numbers.

b) Prove that there are infinitely many Ulam numbers.

What are the values of these sums?

$\begin{array}{r}\sum _{k=1}^5 (k+1)\\ \sum _{j=0}^4 (-2{)}^j\\ \sum _{j=1}^{10} 3\\ \sum _{j=0}^8 \left(2^{j+1}-2^j\right)\end{array}$

Determine whether each of these 15 -digit numbers is a valid airline ticket identification number.

a) 101333341789013

b) 007862342770445

c) 113273438882531

d) 000122347322871

What is the Cartesian product A X B X C, where A is the set of all airlines and B and C are both the set of all cities in the United states? Give an example of how this Cartesian product can be used.

Show that the function $f\left(x\right)=\left|x\right|$ 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.

What can you say about the set A and B if we know that

a)\(A \cup B = A?\)

b)\(A \cap B = A?\)

c)\(A - B = A?\)

d)\(A \cap B = B \cap A?\)

e)\(A - B = B - A?\)