Mathematics is the indispensable instrument of

all physical research. - BERTHELOT

2.1 Introduction

Much of mathematics is about finding a pattern - a recognisable link between quantities that change. In our daily life, we come across many patterns that characterise relations such as brother and sister, father and son, teacher and student. In mathematics also, we come across many relations such as number $m$ is less than number $n$, line $l$ is parallel to line $m$, set $A$ is a subset of set $B$. In all these, we notice that a relation involves pairs of objects in certain order. In this Chapter, we will learn how to link pairs of objects from two sets and then introduce relations between the two objects in the pair. Finally, we will learn about special relations which will qualify to be functions. The

concept of function is very important in mathematics since it captures the idea of a mathematically precise correspondence between one quantity with the other.

2.2 Cartesian Products of Sets

Suppose A is a set of 2 colours and B is a set of 3 objects, i.e.,

$$ A=\{\text { red, blue }\} \text { and } B=\{b, c, s\} \text {, } $$

where $b, c$ and $s$ represent a particular bag, coat and shirt, respectively.

How many pairs of coloured objects can be made from these two sets?

Proceeding in a very orderly manner, we can see that there will be 6 distinct pairs as given below:

(red, $b$ ), (red, $c$ ), (red, $s$ ), (blue, $b$ ), (blue, $c$ ), (blue, $s$ ).

Thus, we get 6 distinct objects (Fig 2.1).

Let us recall from our earlier classes that an ordered pair of elements taken from any two sets $P$ and $Q$ is a pair of elements written in small brackets and grouped together in a particular order, i.e., $(p, q), p \in P$ and $q \in Q$. This leads to the following definition:

Definition 1 Given two non-empty sets $P$ and $Q$. The cartesian product $P \times Q$ is the set of all ordered pairs of elements from $P$ and $Q$, i.e.,

$$ P \times Q=\{(p, q): p \in P, q \in Q\} $$

If either $P$ or $Q$ is the null set, then $P \times Q$ will also be empty set, i.e., $P \times Q=\phi$

From the illustration given above we note that

$A \times B=\{(red, b),($ red,$c),($ red,$s),($ blue,$b),($ blue,$c),($ blue,$s)\}$.

Again, consider the two sets:

$A=\{DL, MP, KA\}$, where DL, MP, KA represent Delhi, Madhya Pradesh and Karnataka, respectively and B $=\{01,02, 03 \}$ representing codes for the licence plates of vehicles issued by DL, MP and KA.

If the three states, Delhi, Madhya Pradesh and Karnataka were making codes for the licence plates of vehicles, with the restriction that the code begins with an element from set $A$, which are the pairs available from these sets and how many such pairs will there be (Fig 2.2)?

The available pairs are:(DL,01), (DL,02), (DL,03), (MP,01), (MP,02), (MP,03), $(KA, 01),(KA, 02),(KA, 03)$ and the product of set $A$ and set $B$ is given by $A \times B=\{(DL, 01),(DL, 02),(DL, 03),(MP, 01),(MP, 02),(MP, 03),(KA, 01),(KA, 02)$, $(KA, 03)\}$.

It can easily be seen that there will be 9 such pairs in the Cartesian product, since there are 3 elements in each of the sets A and B. This gives us 9 possible codes. Also note that the order in which these elements are paired is crucial. For example, the code (DL, 01 ) will not be the same as the code $(01, DL)$.

As a final illustration, consider the two sets $A=\{a_1, a_2\}$ and

$B=\{b_1, b_2, b_3, b_4\}$ (Fig 2.3).

$A \times B=\{(a_1, b_1),(a_1, b_2),(a_1, b_3),(a_1, b_4),(a_2, b_1),(a_2, b_2),(a_2, b_3),(a_2, b_4)\} .$

The 8 ordered pairs thus formed can represent the position of points in the plane if A and B are subsets of the set of real numbers and it is obvious that the point in the position $(a_1, b_2)$ will be distinct from the point in the position $(b_2, a_1)$.

Remarks

(i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal.

(ii) If there are $p$ elements in $A$ and $q$ elements in $B$, then there will be $p q$ elements in $A \times B$, i.e., if $n(A)=p$ and $n(B)=q$, then $n(A \times B)=p q$.

(iii) If $A$ and $B$ are non-empty sets and either $A$ or $B$ is an infinite set, then so is $A \times B$.

(iv) $A \times A \times A=\{(a, b, c): a, b, c \in A\}$. Here $(a, b, c)$ is called an ordered triplet.

2.1 Relations

Consider the two sets $P=\{a, b, c\}$ and $Q=\{$ Ali, Bhanu, Binoy, Chandra, Divya $\}$.

The cartesian product of $P$ and $Q$ has 15 ordered pairs which can be listed as $P \times Q=\{(a, \text{Ali})$, (a, Bhanu), (a, Binoy), …, (c, Divya) $\}$.

We can now obtain a subset of $P \times Q$ by introducing a relation $R$ between the first element $x$ and the second element $y$ of each ordered pair $(x, y)$ as

$R=\{(x, y): x$ is the first letter of the name $y, x \in P, y \in Q\}$.

Then $R=\{(a, Ali),(b, Bhanu),(b, Binoy),(c$, Chandra $)\}$

A visual representation of this relation $R$ (called an arrow diagram) is shown in Fig 2.4.

Definition 2 A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of the cartesian product $A \times B$. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in $A \times B$. The second element is called the image of the first element.

Definition 3 The set of all first elements of the ordered pairs in a relation $R$ from a set A to a set $B$ is called the domain of the relation $R$.

Definition 4 The set of all second elements in a relation $R$ from a set $A$ to a set $B$ is called the range of the relation $R$. The whole set $B$ is called the codomain of the relation $R$. Note that range $\subset$ codomain.

Remarks (i) A relation may be represented algebraically either by the Roster method or by the Set-builder method.

(ii) An arrow diagram is a visual representation of a relation.

2.4 Functions

In this Section, we study a special type of relation called function. It is one of the most important concepts in mathematics. We can, visualise a function as a rule, which produces new elements out of some given elements. There are many terms such as ‘map’ or ‘mapping’ used to denote a function.

Definition 5 A relation $f$ from a set $A$ to a set $B$ is said to be a function if every element of set $A$ has one and only one image in set $B$.

In other words, a function $f$ is a relation from a non-empty set $A$ to a non-empty set $B$ such that the domain of $f$ is $A$ and no two distinct ordered pairs in $f$ have the same first element.

If $f$ is a function from A to B and $(a, b) \in f$, then $f(a)=b$, where $b$ is called the image of $a$ under $f$ and $a$ is called the preimage of $b$ under $f$.

The function $f$ from $A$ to $B$ is denoted by $f: A \rightarrow B$.

Looking at the previous examples, we can easily see that the relation in

2.4.1 Some functions and their graphs

(i) Identity function Let $\mathbf{R}$ be the set of real numbers. Define the real valued function $f: \mathbf{R} \rightarrow \mathbf{R}$ by $y=f(x)=x$ for each $x \in \mathbf{R}$. Such a function is called the identity function. Here the domain and range of $f$ are $\mathbf{R}$. The graph is a straight line as shown in Fig 2.8. It passes through the origin.

(ii) Constant function Define the function $f: \mathbf{R} \rightarrow \mathbf{R}$ by $y=f(x)=c, x \in \mathbf{R}$ where $c$ is a constant and each $x \in \mathbf{R}$. Here domain of $f$ is $\mathbf{R}$ and its range is $\{c\}$.

The graph is a line parallel to $x$-axis. For example, if $f(x)=3$ for each $x \in \mathbf{R}$, then its graph will be a line as shown in the Fig 2.9.

(iii) Polynomial function A function $f: \mathbf{R} \rightarrow \mathbf{R}$ is said to be polynomial function if for each $x$ in $\mathbf{R}, y=f(x)=a_0+a_1 x+a_2 x^{2}+\ldots+a_{n} x^{n}$, where $n$ is a non-negative integer and $a_0, a_1, a_2, \ldots, a_{n} \in \mathbf{R}$.

The functions defined by $f(x)=x^{3}-x^{2}+2$, and $g(x)=x^{4}+\sqrt{2} x$ are some examples

of polynomial functions, whereas the function $h$ defined by $h(x)=x^{\frac{2}{3}}+2 x$ is not a polynomial function.(Why?)

2.4.2 Algebra of real functions

In this Section, we shall learn how to add two real functions, subtract a real function from another, multiply a real function by a scalar (here by a scalar we mean a real number), multiply two real functions and divide one real function by another.

(i) Addition of two real functions Let $f: X \rightarrow \mathbf{R}$ and $g: X \rightarrow \mathbf{R}$ be any two real functions, where $X \subset \mathbf{R}$. Then, we define $(f+g): X \rightarrow \mathbf{R}$ by

$(f+g)(x)=f(x)+g(x)$, for all $x \in \mathbf{X}$.

(ii) Subtraction of a real function from another $Let f: X \rightarrow \mathbf{R}$ and $g: X \rightarrow \mathbf{R}$ be any two real functions, where $\mathbf{X} \subset \mathbf{R}$. Then, we define $(f-g): X \rightarrow \mathbf{R}$ by $(f-g)(x)=f(x)-g(x)$, for all $x \in X$.

(iii) Multiplication by a scalar Let $f: X \rightarrow \mathbf{R}$ be a real valued function and $\alpha$ be a scalar. Here by scalar, we mean a real number. Then the product $\alpha f$ is a function from $X$ to $\mathbf{R}$ defined by $(\alpha f)(x)=\alpha f(x), x \in X$.

(iv) Multiplication of two real functions The product (or multiplication) of two real functions $f: \mathbf{X} \rightarrow \mathbf{R}$ and $g: X \rightarrow \mathbf{R}$ is a function $f g: X \rightarrow \mathbf{R}$ defined by $(f g)(x)=f(x) g(x)$, for all $x \in X$.

This is also called pointwise multiplication.

(v) Quotient of two real functions Let $f$ and $g$ be two real functions defined from $X \rightarrow \mathbf{R}$, where $X \subset \mathbf{R}$. The quotient of $f$ by $g$ denoted by $\frac{f}{g}$ is a function defined by , $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$, provided $g(x) \neq 0, x \in X$

Summary

In this Chapter, we studied about relations and functions. The main features of this Chapter are as follows:

Ordered pair A pair of elements grouped together in a particular order.

Cartesian product $A \times B$ of two sets $A$ and $B$ is given by

$A \times B=\{(a, b): a \in A, b \in B\}$

In particular $\mathbf{R} \times \mathbf{R}=\{(x, y): x, y \in \mathbf{R}\}$

and $\mathbf{R} \times \mathbf{R} \times \mathbf{R}=\{(x, y, z): x, y, z \in \mathbf{R}\}$

If $(a, b)=(x, y)$, then $a=x$ and $b=y$. If $n(A)=p$ and $n(B)=q$, then $n(A \times B)=p q$.

$\Delta A \times \phi=\phi$

In general, $A \times B \neq B \times A$.

Relation A relation $R$ from a set $A$ to a set $B$ is a subset of the cartesian product $A \times B$ obtained by describing a relationship between the first element $x$ and the second element $y$ of the ordered pairs in $A \times B$.

The image of an element $x$ under a relation $R$ is given by $y$, where $(x, y) \in R$,

The domain of $R$ is the set of all first elements of the ordered pairs in a relation $R$.

The range of the relation $R$ is the set of all second elements of the ordered pairs in a relation $R$.

Function A function $f$ from a set $A$ to a set $B$ is a specific type of relation for which every element $x$ of set $A$ has one and only one image $y$ in set $B$.

We write $f: A \rightarrow B$, where $f(x)=y$.

A is the domain and B is the codomain of $f$.

The range of the function is the set of images.

A real function has the set of real numbers or one of its subsets both as its domain and as its range.

Algebra of functions For functions $f: X \rightarrow \mathbf{R}$ and $g: X \rightarrow \mathbf{R}$, we have

$$ \begin{aligned} & (f+g)(x)=f(x)+g(x), x \in X \\ & (f-g)(x)=f(x)-g(x), x \in X \\ & (f . g)(x) \quad=f(x) \cdot g(x), x \in X \\ & (k f)(x) \quad=k(f(x)), x \in X, \text { where } k \text { is a real number. } \\ & (\frac{f}{g})(x)=\frac{f(x)}{g(x)}, x \in X, g(x) \neq 0 \end{aligned} $$

Historical Note

The word FUNCTION first appears in a Latin manuscript “Methodus tangentium inversa, seu de fuctionibus” written by Gottfried Wilhelm Leibnitz (1646-1716) in 1673; Leibnitz used the word in the non-analytical sense. He considered a function in terms of “mathematical job” - the “employee” being just a curve.

On July 5, 1698, Johan Bernoulli, in a letter to Leibnitz, for the first time deliberately assigned a specialised use of the term function in the analytical sense. At the end of that month, Leibnitz replied showing his approval.

Function is found in English in 1779 in Chambers’ Cyclopaedia: “The term function is used in algebra, for an analytical expression any way compounded of a variable quantity, and of numbers, or constant quantities”.