114 Examples

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Here are some examples of the definitions and theorems given on the 114 page.

1 Mappings and Functions
- 1.1 Definitions and Properties
- 1.2 Operations
2 Groups and Permutations
- 2.1 Introduction to Groups
- 2.2 Permutations

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Mappings and Functions

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Definitions and Properties

The following are mappings:

$f : \mathbb{R}\rightarrow \mathbb{R}$ , $f(x) = x^2$ for all $x\in\mathbb{R}$ .
$g : \mathbb{Z}\rightarrow \mathbb{R}$ , $g(x) = e^{x+1}$ for all $x\in\mathbb{Z}$ .
$h : \mathbb{C}\rightarrow \mathbb{R}$ , $h(a+bi) = \sqrt{a^2+b^2}$ for all $a,b\in\mathbb{R}$ .

The mappings $f : \mathbb{R}\rightarrow \mathbb{R}$ and $g : \mathbb{R}\rightarrow \mathbb{R}$ given by $f(x) = x+1$ and $g(x) = \frac{x^2+x+1}{x+1} + \frac{x}{x+1}$ are not equal. This is because, although if we simplify $g$ we will get $f$ , the mapping $g$ cannot be defined at $x = -1$ , and therefore the mappings do not have the same domain. If we define $g(-1) = 0$ , then the mappings are equal.

The following mappings are injective:

$f(x) = x$
$g(x) = 2x+1$
$h(x) = \frac{e^x}{4}$

The following mappings are not injective:

$f(x) = x^2$ , as $f(-1) = f(1)$ , but $1\neq -1$
$g(x) = (\sin x)^2$ , as $g(0) = g(2\pi)$ , but $0 \neq 2\pi$

The following mappings are surjective:

$f(x) = x$ , by definition.
$g(x) = 2x+1$ , as $g\left(\frac{x-1}{2}\right) = x$ .
$h(x) = \frac{e^x}{4}$ , as $h(\ln 4x) = x$ .

The following mappings are not surjective:

$f : \mathbb{R} \rightarrow \mathbb{R}$ , $f(x) = \frac{1}{x}$ , as the does not exist a real number such that $\frac{1}{x} = 0$

Note that the above three functions that were both one to one and onto are bijective.

Let $f : \mathbb{Z}\rightarrow\mathbb{Q}$ and $g : \mathbb{Q}\rightarrow \mathbb{R}$ be defined by $f(n) = \frac{n}{n+1}$ where $f(-1) = 0$ , and $g(x) = 2e^x$ . Thus, if we have $h = g\circ f$ , then $h(n) = g(f(n)) = 2e^{\frac{n}{n+1}}$ , where $h(-1) = 2$ .

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Operations

Given the set $\mathbb{N}$ , the relation $n*m = n^m$ is an operation on that set.

We have seen many examples of abridged forms of Cayley tables before. Consider for example:

×	1	2	3	4	5
1	1	2	3	4	5
2	2	4	6	8	10
3	3	6	9	12	15
4	4	8	12	16	20
5	5	10	15	20	25

The above is a section of the Cayley table for the natural numbers under the operation of multiplication.

We know that $a+(b+c) = (a+b)+c$ for all real numbers, thus addition is associative on $\mathbb{R}$ . However, the operation $*$ defined by $n*m = n^m$ is not associative, as $2*(3*2) = 2^9 = 512$ but $(2*3)*2 = 8^2 = 64$ .

We know that $a+b = b+a$ for all real numbers, thus addition is commutative on $\mathbb{R}$ . However, the operation $*$ defined by $n*m = n^m$ is not commutative, as $2*3 = 8$ but $3*2 = 9$ .

Given the set $\mathbb{Z}$ under addition, we know that for all $a\in\mathbb{Z}$ , $0+a = a+0 = a$ . Thus, $0$ is the identity for addition. Furthermore, given any $a\in\mathbb{Z}$ , $a+(-a) = (-a)+a = 0$ , and thus $-a$ is the inverse of $a$ .

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Groups and Permutations

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Introduction to Groups

The following are groups:

$(\mathbb{Z}, +)$
$(\mathbb{Q}, \cdot)$
The set $G = \left\lbrace 1, -1 \right\rbrace$ under multiplication.
The set of all matrices under matrix addition.
The set of formal pairs $E-E^{\prime}$ of vector bundles over a compact Hausdorff space $X$ where $E_1 - E_1^{\prime} = E_2 - E_2^{\prime}$ if $E_1 \oplus E_2^{\prime}$ is stably isomorphic to $E_2 \oplus E_1^{\prime}$ (where two bundles $E$ and $E^{\prime}$ are stably isomorphic if they become isomorphic upon the addition of a trivial bundle, $E\oplus\varepsilon^n \approx E^{\prime} \oplus \varepsilon^n$ ).

The following are not:

$(\mathbb{N}, +)$ , because inverses do not exist.
$(\mathbb{Z}, \cdot)$ because inverses do not exist.
$(\mathbb{Q}, *)$ where $n*m = n^m$ , as the operation is not associative.
The set of all matrices under matrix multiplication.

Note that all groups are also semi-groups. Also, $(\mathbb{N}, +)$ and $(\mathbb{N}, \cdot)$ are also semi-groups.

Also note that as addition and multiplication are commutative, we know that all the abvoe groups whose operations are eithe addition or multiplication are Abelian groups.

The order of all of the groups involving $\mathbb{Z}$ or $\mathbb{Q}$ are infinite. However the third group above has order 2.

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Permutations

The elements of $S_3$ are given by:

$\begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix}$	$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix}$	$\begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{pmatrix}$
$\begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix}$	$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}$	$\begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{pmatrix}$

Writen more compactly, we have $S_3 = \left\lbrace \begin{pmatrix} 1 \end{pmatrix}, \begin{pmatrix} 1 & 2 \end{pmatrix}, \begin{pmatrix} 1 & 3 \end{pmatrix}, \begin{pmatrix} 2 & 3 \end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \end{pmatrix}, \begin{pmatrix} 1 & 3 & 2 \end{pmatrix}, \right\rbrace$