Ordered Pair

Two elements $a \ and \ b$ listed in a specific order, form an ordered pair, denoted by $a \ and \ b$.

In an ordered pair $a \ and \ b$, we call a as the first component and b as the second component.

By changing the positions of the components, the ordered pair is changed, i.e. $(5, \ 4) \ne (4, \ 5)$

Also if $(a, \ b) = (c, d) \Rightarrow a=c \ and \ b=d$

Thus $(a, \ b) = (2, 7) \Rightarrow a=2 \ and \ b=7$ $\\$

Cartesian Product of Two Sets

Let $A \ and \ B$ be two non-empty sets. Then their Cartesian product $A \ \times \ B$  is the set of all ordered pairs $a \ and \ b$ such that $a \in A \ and\ b\in B.$ $\\$

Example 1:

Let $A=\{a,\ b \} \ and \ B=\{1, \ 2, \ 3\}$

Then $A \times B = \{a,\ b \} \times \{1, \ 2, \ 3\} = \{ (a, 1), \ (a, 2),\ (a, 3),\ (b, 1),\ (b, 2),\ (b, 3)\}$ $\\$

Relation

Let $A \ and \ B$  be two non-empty sets. Then every subset of $A \times B$ is called a relation from $A \ to\ B$. i.e., if $R \subseteq A\times B$, then $R$ is a relation from $A \ to \ B$ $\\$

Representation of a Relation

Roster Form:  When a relation is represented by the set of all ordered pairs contained in it, then it is said to be in roster form. $\\$

Example 2:

If $A=\{1, 9, 16, 25\} \ and\ B=\{1, 2, 3, 4, 5\}$  then a relation $R \ from\ A \ to\ B$   defined as ‘is the square of’ can be represented in the roster form as: $R = \{(1,1), (9,3), (16, 4), (25, 5)\}$

You will notice that the first component of each subset is square of the second component of the subset. $\\$ Arrow Diagram: Let $R$ be a relation from $A \ to \ B$. We can draw the sets as shown pictorially.  Then, we draw arrows from $A \ to \ B$ to indicate the pairing of the corresponding elements related to each other. Thus, we can show the relation given in Example 2 by the arrow diagram as shown. $\\$

Set Builder Form: A relation $R$ from $A \ to \ B$ is said to be in a Set-Builder form when written as: $R=\{(a,\ b): a \in A,\ b \in B\ and\ a \ is \ connected \ with \ b \ by \ a \ given rule\}$

The relation given in Example 2 can be represented in the set-builder form as: $R=\{(a,\ b): a \in A,\ b \in B\ and\ a=b^2\}$ $\\$

Domain and Range of a Relation

Let $R$ be a relation from $A \ to \ B$. Then,

Domain ( $R$ ) = Set of first components of all ordered pairs in $R$

Range ( $R$ ) = Set of second components of all ordered pairs in $R$ $\\$

Rule of Relation

The property that tells us how the first component is related to the second component of each ordered pair in $R$, is called the rule of the relation.

Let’s look at the following example: $\\$

Example 3:

Let $A=\{2,\ 3,\ 4\} \ and\ B=\{1, \ 3, \ 5\}$

Let $R$ be the relation ‘is less than’ from $A \ to \ B$. Find $R$. Also, write down the domain and range of $R$.

Solution:

We have: $R=\{(a,\ b): a \in A,\ b \in B\ and\ a < b \} = \{(2,\ 3),\ (2,\ 5),\ (3,\ 5),\ (4,\ 5) \}$

Note: We take only those pairs $(a, \ b) \ of A\times B$ in which $a < b$

Hence, $R = \{(2,3), \ (2,5), \ (4,5) \}$

Domain ( $R$ ) = Set of first components of elements of $R = \{2,\ 3,\ 4\}$

Range ( $R$ ) = Set of second components of elements of $R = \{3,\ 5\}$