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

32. Express the negations of each of these statements so thatall negation symbols immediately precede predicates.
a) \(\exists \user1{z}\forall \user1{y}\forall \user1{xT}\left( {\user1{x, y, z}} \right)\)
b)\(\exists \user1{x}\exists \user1{yP}\left( {\user1{x,y}} \right) \wedge \forall \user1{x}\forall \user1{yQ}\left( {\user1{x, y}} \right)\)
c)\(\exists \user1{x}\exists \user1{y}\left( {\user1{Q}\left( {\user1{x,y}} \right) \leftrightarrow \user1{Q}\left( {\user1{y, x}} \right)} \right)\)
d) \(\forall \user1{y}\exists \user1{x}\exists \user1{z}\left( {\user1{T}\left( {\user1{x,y,z}} \right) \vee \user1{Q}\left( {\user1{x, y}} \right)} \right)\)

Find the value of each of these sums.

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

Question: let$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathit{x}$ where the domain is the set of real numbers. What is

1. $\mathit{f}\mathbf{\left(}\mathit{Z}\mathbf{\right)}$?
2. $\mathit{f}\mathbf{\left(}\mathit{N}\mathbf{\right)}$?
3. $\mathit{f}\mathbf{\left(}\mathit{R}\mathbf{\right)}$?

let $f\left(x\right)=2x$ where the domain is the set of real numbers. What is

1. $f\left(Z\right)$?
2. $f\left(N\right)$?
3. $f\left(R\right)$?

Find the symmetric difference of\(A = \left\{ {1,3,5} \right\}\)and\(B = \left\{ {1,2,3} \right\}\).

In this exercise we show that the meet and join operations are associative. Let A,B and C bezero-one matrices. Show that

a) $\mathbf{(}\mathit{A}\mathbf{\vee }\mathit{B}\mathbf{)}\mathbf{\vee }\mathit{C}\mathbf{=}\mathit{A}\mathbf{\vee }\mathbf{(}\mathit{B}\mathbf{\vee }\mathit{C}\mathbf{)}$ b)$\mathbf{(}\mathit{A}\mathbf{\wedge }\mathit{B}\mathbf{)}\mathbf{\wedge }\mathit{C}\mathbf{=}\mathit{A}\mathbf{\wedge }\mathbf{(}\mathit{B}\mathbf{\wedge }\mathit{C}\mathbf{)}$

Let \({\bf{A = }}\left\{ {{\bf{a,b,c}}} \right\}\), \({\bf{B = }}\left\{ {{\bf{x,y}}} \right\}\) and \({\bf{C = }}\left\{ {{\bf{0,1}}} \right\}\) , Find.

(a) \({\bf{A \times B \times C}}\)

(b) \({\bf{C \times B \times A}}\)

(c)\({\bf{C \times A \times B}}\)

(d) \({\bf{B \times B \times B}}\)

Show that when you substitute ${\left(3n+1\right)}^2$for each occurrence of n and ${\left(3m+1\right)}^2$for each occurrence of m in the right-hand side of the formula for the function $f\left(m,n\right)$in Exercise 31 , you obtain a one-to-one polynomial function $Z\times Z\to Z$. It is an open question whether there is a one-to-one polynomial function $Q\times Q\to Q$ .

Show that the set of irrational numbers is an uncountable set.

Are each of these eight-digit codes possible ISSNs? That is, do they end with a correct check digit?

a) 1059-1027

b) 0002-9890

c) 1530-8669

d) 1007-120 x