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

Let $\mathit{A}\mathbf{=}\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]\mathbf{\text{and}}\mathit{B}\mathbf{=}\left[\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]$

Find

a) $\mathit{A}\mathbf{\vee }\mathit{B}$.

b) $\mathit{A}\mathbf{\wedge }\mathit{B}$.

c)$\mathit{A}\mathbf{\odot }\mathit{B}$.

Show that the union of a countable number of countable sets is countable.

Question: a) Prove that a strictly decreasing function from R to itself is one-to-one.

b) Give an example of a decreasing function from R to itself is not one-to-one.

a) Prove that a strictly decreasing function from R to itself is one-to-one.

b) Give an example of a decreasing function from R to itself is not one-to-one.

Show that \(\left\{ {{1^p}\mid p\;\;\;prime} \right\}\} is not regular. You may use the pumping lemma given in Exercise 22 of Section 13.4.

Prove that if m is a positive integer and x is a real number, then

$\lfloor mx\rfloor =\lfloor x\rfloor +⌊x+\frac{1}{m}⌋+⌊x+\frac{2}{m}⌋+\dots +⌊x+\frac{m-1}{m}⌋$

Let$A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$

Find

a) ${\mathit{A}}^{\left[2\right]}$.

b)${\mathit{A}}^{\left[3\right]}$

c) $\mathit{A}\mathbf{\vee }{\mathit{A}}^{\mathbf{\left[}\mathbf{2}\mathbf{\right]}}\mathbf{\vee }{\mathit{A}}^{\mathbf{\left[}\mathbf{3}\mathbf{\right]}}$.

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

Let an be the nth term of the sequence

$1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,\dots$constructed by including the integer exactly times. Show that${\mathrm{a}}_{\mathrm{n}}=\left[\sqrt{2\mathrm{n}}+\frac{1}{2}\right.⌋$

Draw the Venn diagrams for each of these combinations of sets A, B, C, and D.

a) \(\left( {A \cap B} \right) \cup \left( {C \cap D} \right)\)

b) \(\mathop A\limits^\_ \cup \mathop B\limits^\_ \cup \mathop C\limits^\_ \cup \mathop D\limits^\_ \)

c) \(A - \left( {B \cap C \cap D} \right)\)