Ordered Pairs

Ordered Pairs:  An ordered pair consists of two objects or elements in a given fixed order.

For example, if $A$ and $B$ are any two sets, then by an ordered pair of elements we mean a pair $(a,b)$ in that order, where $a \in A$, $b \in B$.

In the ordered pair $(a,b)$, $a$ is called the first component (coordinate) and $b$ is called the second component (coordinate).

Note: An ordered pair is not a set consisting of two elements. The ordering of the two elements in an ordered pair is important and the two elements need not be distinct.

Equality Of Ordered Pairs: Two ordered pairs $(a_1,b_1)$ and $(a_2,b_2)$ are equal iff $a_1=a_2$ and $b_1=b_2$.

i.e $(a_1, b_1) = (a_2, b_2) \Leftrightarrow a_1 = a_2$ and $b_1 = b_2$.

It is evident from this definition that $(1,2) \neq (2,1)$ and $(1, 1) \neq (2,2)$.

Cartesian Product Of Sets

Cartesian Product Of Sets:  Let $A$ and $B$ be any two non-empty sets. The set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$ is called the Cartesian product of the sets $A$ and $B$ and is denoted by $A \times B$.

Thus, $A \times B = \{ (a,b) : a \in A \ and \ b \in B \}$. If $A=\phi$ or $B=\phi$, then we define $A \times B=\phi$

Cartesian Product Of Three Sets: Let $A, B$ and $C$ be three sets.Then, $A \times B \times C$ is the set of all ordered triplets having first element from $A$, second element from $B$ and third element from $C$.

i.e. $A \times B \times C = \{ (a,b,c) : a \in A, b \in B, c \in C \}$

Note: It should be noted that $A \times B \times C = (A \times B) \times C = A \times (B \times C)$

Number Of Elements In The Cartesian Product Of Two Sets

If $A$ and $B$ are two finite sets, then $n(A \times B) = n(A) \times n(B)$

Note:  (i) If either $A$ or $B$ is an infinite set, then $A \times B$ is an infinite set.  (ii) If $A,B,C$ are finite sets, then $n(A \times B \times C) = n(A) \times n(B) \times n(C)$

Graphical Representation Of Cartesian Product Of Sets

Let $A$ and $B$ be any two non-empty sets. To represent $A \times B$ graphically, we draw two mutually perpendicular lines, one horizontal and other vertical. On the horizontal line, we represent the elements of set $A$ and on the vertical line, the elements of $B$. If $a \in A$, $b \in B$, we draw a vertical line through $a$ and a  horizontal line through $b$.
These two lines will meet in a point which will denote the ordered pair $(a, b)$. In this manner we mark points corresponding to each ordered pair in $A \times B$.

For Example,  If $A = \{ 1, 2, 3 \}$ and $B = \{ 2, 4 \}$ , then, $A \times B = \{ (1, 2), (1, 4), (2,2), (2, 4), (3,2),(3, 4) \}$.

Diagrammatic Representation Of Cartesian Product Of Two Sets

In order to represent $A \times B$ by an arrow diagram, we first draw Venn diagrams representing sets $A$ and $B$ one opposite to the other as shown in adjoining figure. Now, we draw line segments starting from each element of $A$ and terminating to each element of set $B$. If $A = \{ 1, 3, 5 \}$ and $B = \{ a, b \}$, then the arrow diagram of $A \times B$ is shown in the figure.

Relations

Relation: Let $A$ and $B$ be two sets. Then a relation $R$ from $A$ to $B$ is a subset of $A \times B$. Thus, $R$ is a relation from $A$ to $B \Leftrightarrow R \subseteq A \times B$. If $R$ is a relation from a non-void set $A$ to a non-void set $B$ and if $(a,b) \in R$, then we write $a \ R \ b$ which is read as , “$a$ is related to $b$ by the relation $R$” . If $(a, b) \notin R$, then we write $a \$ R $\ b$ and we say that a is not related to $b$ by the relation $R$.

Total Number of Relations: Let $A$ and $B$ be two non-empty finite sets consisting of $m$ and $n$ elements respectively. Then $A \times B$ consists of $mn$ ordered pairs. So, total number of subsets of $A \times B$ is $2^{mn}$. Since each subset of $A \times B$ defines a relation from $A$ to $B$, so total number of relations from $A$ to $B$ is $2^{mn}$. Among these $2^{mn}$ relations the void relation $\phi$ and the universal relation $A \times B$ are trivial relations from $A$ to $B$.

Representation of a Relation

A relation from a set $A$ to a set $B$ can be represented in, any one of the following forms:

Roster Form: In this form a relation is represented by the set of all ordered pairs belonging to $R$.

For example, if  $R$ is a relation from set $A = \{ -2,-1, \ldots ,0,1,2 \}$ to set $B=\{ 0, 1, 4, 9, 10 \}$ by the rule $a R b \Leftrightarrow a^2 =b$. Then, $0 \ R \ 0, -2 \ R \ 4, -1 \ R \ 1, 1 \ R \ 1$ and $2 \ R \ 4$ .

So, $R$ can be described in Roster form as follows: $R = \{ (0, 0), (- 1, 1), (- 2, 4), (1, 1), (2, 4) \}$

Set Builder Form:  In this form the relation $R$ from set $A$ to set $B$ is represented as
$R = \{ (a, b): a \in A, b \in B \ and \ a,b \ satisfy \ the \ rule \ which \ associates \ a \ and \ b \}$

For example, if $A=\{ 1, 2, 3, 4, 5 \}$ and $B = \{ 1,$ $\frac{1}{2}$ $,$ $\frac{1}{3}$ $,$ $\frac{1}{4}$ $,$ $\frac{1}{5}$ $,$ $\frac{1}{6}$ $,$ $\ldots \}$ and $R$ is a relation from $A$ to $B$ given by $R = \{ (1, 1), (2,$ $\frac{1}{2}$ $), (3,$ $\frac{1}{3}$ $), (4,$ $\frac{1}{4}$ $), (5,$ $\frac{1}{5}$ $) \}$

Then $R$ in set-builder form can be described as follows: $R = \{ (a, b) : a \in A , b \in B \ and \ b =$ $\frac{1}{a}$ $\}$

By Arrow Diagram: In order to represent a relation from set $A$ to a set $B$ by an arrow diagram, we draw arrows from first components to the second components of all ordered pairs belonging to $R$.

For example, relation $R =\{ 1,a), (1,b), (3,a), (3, b), (5,a), (5, b) \}$ from set $A = \{ 1,3,5 \}$ to set $B = \{ a, b \}$ can be represented by the arrow diagram shown in figure.

By Lattice:  In this form, the relation $R$ from set $A$ to set $B$ is represented by darkening the dots in the lattice for $A \times B$ which represent the ordered pairs in $R$.

For example, if $R = \{ (1, 2),(2,2),(3,2),(1,4), (2,4),(3,4) \}$ is a relation from set $A = \{ 1,2,3 \}$ to set $B = \{ 2, 4 \}$ , then $R$ can be represented by the lattice shown in the figure.

Domain and Range of a Relation

Let $R$ be a relation from a set $A$ to a set $B$. Then the set of all first components or coordinates of the ordered pairs belonging to $R$ is called the domain of $R$, while the set of all second components or coordinates of the ordered pairs in $R$ is called the range of $R$.

$\text{Thus, Domain } (R) = \{ a: (a,b) \in R \} \text{ and, Range } (R) = \{ b : (a,b) \in R \}$

It is evident from the definition that the domain of a relation from $A$ to $B$ is a subset of $A$ and its range is a subset of $B$. The set $B$ is called the co-domain of relation $R$.

If $A = \{ 1, 3,5,7 \}, B = \{ 2, 4, 6, 8, 10 \}$ and let $R = \{ (1,8), (3, 6), (5, 2),(1, 4) \}$ be a relation from $A$ to $B$ Then, Dom $(R) = \{ 1, 3,5 \}$ and Range $(R) = \{ 8,6,2, 4 \}$.

Relation on a Set: Let $A$ be a non-void  set. Then, a relation from $A$ to itself i.e. a subset of $A \times A$, is called a relation on set $A$.

Inverse of a Relations: Let $A$ and $B$ be two sets and let $R$ be a relation from a set $A$ to a set $B$. Then the inverse of $R$, denoted by $R^{-1}$, is a relation from $B$ to $A$ and is defined by $R = \{ (b, a): (a, b) \in R \}$

$\text{Clearly, } (a,b) \in R \Rightarrow (b, a) \in R^{-1} \\ \\ \text{Also, Dom } (R) = \text{ Range } (R^{-1}) \text{ and Range } (R) = \text{ Dom } (R^{-1})$

For example, Let $A = \{ 1, 2, 3 \} B = \{ a, b, c, d \}$ be two sets and let $R = \{ (1, a), (1,c), (2, d), (2, c) \}$ be a relation from $A$ to $B$. Then $R^{-1} = \{ (a, 1), (c, 1), (d, 2), (c, 2) \}$ is a relation from $B$ to $A$

$\text{Also, Dom } (R) = \{1,2 \} = \text{ Range } (R^{-1}) \text{ , and Range } (R) = {a, c, d} = \text{ Domain } (R^{-1})$.