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

what is the empty set? Show that the empty set is a subset of every set.

Show that if A is a subset of B, then the power set of A is a subset of the power set of B.

Suppose that \(A \times B = \phi \) , where A and B are sets. What can you conclude?

What are the values of these sums, where S = {1, 3, 5, 7} ?

$\begin{array}{r}\sum _{j\in S} j\\ \sum _{j\in S} j^2\\ \sum _{j\in s} \frac{1}{j}\mid \\ \sum _{j\in S} 1\end{array}$

can u conclude that \(A = B\) if A, B and C are sets such that

a) \(A \cup C = B \cup C\) ? b) \(A \cap C = B \cap C\) ?

c) \(A \cup C = B \cup C\) and \(A \cap C = B \cap C\)?

Let $\mathrm{S}=\{-1,0,2,4.7\}$. Find f(S) if

1. role="math" localid="1668419106110" $f\left(x\right)=1$
2. role="math" localid="1668419096298" $f\left(x\right)=2+1$
   
3. role="math" localid="1668419089300" $f\left(x\right)=[x/5]$
   
4. $f\left(x\right)=[\left(x^2+1\right)/3]$
   

Question: Let $\mathit{S}\mathbf{=}\left\{-1,0,2,4,7\right\}$. Find if $\mathit{f}\mathbf{\left(}\mathit{S}\mathbf{\right)}$

$$\begin{gathered} \mathbf{\text{a)f(x)=1}} \\ \mathbf{\text{b)f(x)=2x+1}} \\ \mathbf{\text{c)}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\lceil x/5\rceil \\ \mathbf{\text{d)}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\lceil (x^2+1)/3\rceil \end{gathered}$$

Show that the set of real numbers that are solutions of quadratic equations $a{x}^2+bx+c=0$, where a, b and c are integers, is countable.

Let be a zero–one matrix. Show that

a) $\mathit{A}\mathbf{\vee }\mathit{A}\mathbf{=}\mathit{A}$

b)$\mathit{A}\mathbf{\wedge }\mathit{A}\mathbf{=}\mathit{A}$

Determine a rule for generating the terms of the sequence that begins 1, 3, 4,8,15,27,50,92,... and find the next four terms of the sequence.