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

Let A be a set. Show that \(\phi \times A = A \times \phi = \phi \).

Let A and B be subsets of a universal set U. Show that \(A \subseteq B\) if and only if \(\overline B \subseteq \overline A \).

What is the value of each of these sums of terms of a geometric progression?

$\begin{array}{r}\sum _{\mathrm{j}=0}^8 \left(3-2^{\mathrm{j}}\right)\\ \sum _{\mathrm{j}=1}^8 2^{\mathrm{j}}\\ \sum _{\mathrm{j}=2}^8 (-3{)}^{\mathrm{j}}\\ \sum _{\mathrm{j}=0}^8 2(-3{)}^{\mathrm{j}}\end{array}$

Show that $Z^{+}\times Z$is countable by showing that the polynomial function $f:Z^{+}\to Z^{+}\to Z^{+}$with $f(m,n)=(m+n-2)(m+n-1)/2+m$ is one-to-one and onto.

Construct a deterministic finite-state automaton that recognizes the set of all bit strings that begin and end with 11.

Show that if \(a \ne {b^d}\) and\(n\)is a power of\(b\), then\(f(n) = {C_1}{n^d} + {C_2}{n^{{{\log }_b}a}}\), where \({C_1} = {b^d}c/\left( {{b^d} - a} \right)\) and\({C_2} = f(1) + {b^d}c/\left( {a - {b^d}} \right)\).

Let $f\left(x\right)=[x^2/3]$. Find f(s) if

1. $S=\{-2,-1,0,1,2\}$
2. $S=\left\{0,1,2,3,4,5\right\}$
   
3. $S=\left\{1,5,7,11\right\}$
   
4. $S=\left\{2,6,10,14\right\}$
   

In this exercise we show that the meet and join operations are commutative. Let A and B be $m\times n$zero-one matrices. Show that

a) $A\vee B=B\vee A$

b)$B\wedge A=A\wedge B$

Question: Let $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\lceil }{\mathbf{x}}^{\mathbf{2}}\mathbf{/}\mathbf{3}\mathbf{\rceil }$. Find $\mathit{f}\mathbf{\left(}\mathit{S}\mathbf{\right)}$if

$$\begin{gathered} \mathbf{\text{a)S={-2,-1,0,1,2,3}}} \\ \mathbf{\text{b)}}\mathbf{S}\mathbf{=\{0,1,2,3}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{5}\mathbf{\}} \\ \mathbf{\text{c)}}\mathbf{S}\mathbf{=\{}\mathbf{1}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{11}\mathbf{\}} \\ \mathbf{\text{d)}}\mathbf{S}\mathbf{=\{}\mathbf{2}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{10}\mathbf{,}\mathbf{14}\mathbf{\}} \end{gathered}$$

Determine a rule for generating the terms of the sequence that begins 2, 3, 3, 5, 10, 133 39,43, 172, 177, 885, 891. ..., and find the next four terms of the sequence.