# Mapping/Examples

## Examples of Mappings

### Rotation of Cartesian Plane Anticlockwise through $30 \degrees$

Let $\Gamma$ be the Cartesian plane.

The rotation $R_{30 \degrees}$ of $\Gamma$ anticlockwise through an angle of $30 \degrees$ about the origin $O$ is a mapping from $\Gamma$ to $\Gamma$.

### $\paren {x - 2}^2 + 1$ Mapping on $\N$

Let $f: \N \to \N$ be the mapping defined on the set of natural numbers as:

- $\forall x \in \N: \map f x = \paren {x - 2}^2 + 1$

Then $f$ has an infinite image set, but is neither a surjection nor an injection.

### $x^3 - x$ Mapping on $\R$

Let $f: \R \to \R$ be the mapping defined on the set of real numbers as:

- $\forall x \in \R: \map f x = x^3 - x$

Then $f$ is a surjection but not an injection.

### Area and Circumference of Circle

Let $A$ denote the set of circles.

Let $f_1: A \to \R$ be the mapping defined on $A$ as:

- $\forall a \in A: \map {f_1} a = \map {\Area} a$

Let $f_2: A \to \R$ be the mapping defined on $A$ as:

- $\forall a \in A: \map {f_2} a = \map {\operatorname {Circ} } a$

where $\map {\operatorname {Circ} } a$ denotes the circumference of $a$.

### Definite Integral

Let $\mathscr C$ be the set of continuous real functions defined on a closed interval $\closedint a b$.

The definite integral is a mapping which associates an element $f \in \mathscr C$ with a real number $\ds \int_a^b \map f x \rd x$.

### Age of a Person

Let $P$ be the set of all people.

Let $\theta: P \to \Z$ be the mapping defined as:

- $\forall x \in P: \map \theta x = \text { the age of $x$ last birthday}$

### Height of a Person

Let $P$ be the set of all people.

Let $\theta: P \to \Z$ be the mapping defined as:

- $\forall x \in P: \map \phi x = \text { the height of $x$ in mm}$

Thus it is seen that multiple mappings may be defined on the same set.

### Population of Nation

Let $T$ be the set of all times.

Let $N$ be a nation in the world.

Let $P: T \to \N$ be the mapping defined as:

- $\forall t \in T: \map P t = \text {the population of $N$ at time $t$}$

### Stretch of Spring

Let $S$ be a spring.

Let $W$ be a weight placed on the end of $S$.

Let $G: \R \to \R$ be the mapping defined as:

- $\map G W = \text {the stretch of $S$ under $W$}$

### Marks in an Examination

The act of marking an examination may be considered as an exercise in creating a mapping from a set of students to a set of numbers in a precisely defined scale, typically between $0$ and $100$.

### Arbitrary Sets of Students

Consider the indexed family as defined in Arbitrary Sets of Students:

Let $S$ be the set of students at a given university.

Let:

- $A_1$ denote the set of first year students
- $A_2$ denote the set of second year students
- $A_3$ denote the set of third year students
- $A_4$ denote the set of fourth year students.

We have:

- $I = \set {1, 2, 3, 4}$ is an indexing set.

Hence $\alpha: I \to S$ is an indexing function on $S$.

Hence:

- $\ds \bigcup_{\alpha \mathop \in I} A_\alpha = $ the set of all undergraduates at the university

and:

- $\ds \bigcap_{\alpha \mathop \in I} A_\alpha = \O$

where $\ds \bigcup_{\alpha \mathop \in I} A_\alpha$ and $\ds \bigcap_{\alpha \mathop \in I} A_\alpha$ denote the union of $\family {A_\alpha}$ and intersection of $\family {A_\alpha}$ respectively.

The indexing function $\alpha: I \to S$ which establishes a correspondence between elements of $I$ and $S$ is an example of a mapping.

## Further Examples of Relations which May or May Not be Mappings

### Questionable mapping on Arbitrary Sets

Let $A = \set {a, b, c}$.

Let $B = \set {1, 2, 3, 4}$.

Let $g_1 \subseteq {A \times B}$ such that:

- $g_1 = \set {\tuple {a, 2}, \tuple {b, 4} }$

Let $g_2 \subseteq {A \times B}$ such that:

- $g_2 = \set {\tuple {a, 3}, \tuple {b, 1}, \tuple {c, 2}, \tuple {c, 4} }$

Let $g_3 \subseteq {A \times B}$ such that:

- $g_3 = \set {\tuple {a, 4}, \tuple {b, 4}, \tuple {c, 2} }$

Then $g_3$ is a mapping.

However, neither $g_1$ nor $g_2$ is a mapping.

The following examples consist of relations on the real numbers $\R$ which may or may not be mappings from $\R$ to $\R$:

### Questionable mapping: $x^2 + y^2 = 1$

Let $R_1$ be the relation defined on the Cartesian plane $\R \times \R$ as:

- $R_1 = \set {\tuple {x, y} \in \R \times \R: x^2 + y^2 = 1}$

Then $R_1$ is not a mapping.

### Questionable mapping: $x y = 1$

Let $R_2$ be the relation defined on the Cartesian plane $\R \times \R$ as:

- $R_2 = \set {\tuple {x, y} \in \R \times \R: x y = 1}$

Then $R_2$ is not a mapping.

### Questionable mapping: $x^4 + y^3 = 1$

Let $R_3$ be the relation defined on the Cartesian plane $\R \times \R$ as:

- $R_3 = \set {\tuple {x, y} \in \R \times \R: x^4 + y^3 = 1}$

Then $R_3$ is a mapping.

### Questionable mapping: $x^3 + y^4 = 1$

Let $R_4$ be the relation defined on the Cartesian plane $\R \times \R$ as:

- $R_4 = \set {\tuple {x, y} \in \R \times \R: x^3 + y^4 = 1}$

Then $R_4$ is not a mapping.

### Questionable mapping: $\sqrt x + \sqrt y = 1$

Let $R_5$ be the relation defined on the Cartesian plane $\R \times \R$ as:

- $R_5 = \set {\tuple {x, y} \in \R \times \R: \sqrt x + \sqrt y = 1}$

Then $R_5$ is not a mapping.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.1$. Mappings: Example $44$