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

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

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

explain what it means for one set to be a subset of another set. How do you prove that one set is a subset of another set?

Let A be the set of English words that contain the letter x, and let B be the set of the English words that contain the letter q. express each of these sets as a combination of A and B.

a) The set of English words that do not contain the letter x.

b) The set of English words that contain both an x and a q.

c) The set of English words that contain an x but not a q.

d) The set of English words that do not contain either an x or a q.

e) The set of English words that contain an x or a q. but not both

Let

$A=\left[\begin{array}{cc}-1& 2\\ 1& 3\end{array}\right]$.

1. Find $A^{-1}$. [Hint: Use Exercise 19.]
2. Find $A^3$.
3. Find ${\left(A^{-1}\right)}^3$.

Use your answers to (b) and (c) to show that ${\left(A^{-1}\right)}^3$is the inverse of $A^3$.

What is the cardinality of each of these sets?

(a) \(\phi \)

(b) \(\left\{ \phi \right\}\)

(c) \(\left\{ {\phi \left\{ \phi \right\}} \right\}\)

(d) \(\left\{ {\phi \left\{ \phi \right\}{\bf{,}}\left\{ {\phi \left\{ \phi \right\}} \right\}} \right\}\)

Assume that the population of the world in 2010 was 6.9 billion and is growing at the rate of 1.1% a year.

a) Set up a recurrence relation for the population of the world n years after 2010.

b) Find an explicit formula for the population of the world n years after 2010.

c) What will the population of the world be in 2030?

Show that if |A| = |B| and |B| = |C| , then |A| = |C| .

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

(a) \(A \cup B = B\)

(b) \(A \cap B = A\)