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

Show that if A,B,C and D are sets |A| = |B| and |C| = |D| then$|\mathrm{A}\times \mathrm{C}|=|\mathrm{B}\times \mathrm{D}|$

Suppose that the number of bacteria in a colony triples every hour.

a) Set up a recurrence relation for the number of bacteria after n hours have elapsed.

b) If 100 bacteria are used to begin a new colony, how many bacteria will be in the colony in 10 hours?

Specify a codomain for each of the functions in Exercise 17. Under what conditions is each of the functions with the codomain you specified onto?

Show that if \(A\) and \(B\) are sets, then

(a) \(\left( {A - B} \right) = A \cap \overline B \)

(b) \(\left( {A \cap B} \right) \cup \left( {A \cap \overline B } \right) = A\)

For which real numbers xand yis it true that(x+y) =

[x] + [y]?

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.

a) the negative integers

b) the even integers

c) the integers less than 100

d) the real numbers between 0 and $\frac{\mathbf{1}}{\mathbf{2}}$

e) the positive integers less than 1,000,000,000

f ) the integers that are multiples of 7

Why is\({\bf{f}}\)not a function from\({\bf{R}}\)to\({\bf{R}}\)if

a)\({\bf{f}}\left( {\bf{x}} \right) = {\bf{1}}/{\bf{x}}?\)

b)\({\bf{f}}\left( {\bf{x}} \right) = \sqrt {\bf{x}} ?\)

c) \({\bf{f}}\left( {\bf{x}} \right) = \pm \sqrt {\left( {{{\bf{x}}^{\bf{2}}} + {\bf{1}}} \right)} ?\)

Let \(A\) be the set of students who live within one mile of school and let \(B\) be the set of students who walk to classes. Describe the students in each of these sets.

(a)\(A \cap B\)

(b)\(A \cup B\)

(c) \(A - B\)

(d) \(B - A\)

$\left[\begin{array}{cccc}1& 1& 1& 3\\ 2& 0& 4& 6\\ 1& 1& 3& 7\end{array}\right]$

a) What size is A?

b) What is the third column of A?

c) What is the second row of A?

d) What is the element of A in the (3, 2)th position?

e) What is role="math" localid="1668434071938" $A^T$?

Determine whether each of these function is \(O(x)\).

a) \(f(x) = 10\)

b)\(f(x) = 3x + 7\)

c)\(f(x) = {x^2} + x + 1\)

d)\(f(x) = 5\log x\)

e)\(f(x) = \left\lfloor x \right\rfloor \)

f)\(f(x) = \left\lceil {x/2} \right\rceil \)