Topic : Relation

📚 Grade 9 | Algebra 📝 Chapter Notes

⚡ Activity : Discovering Relations

Let us consider two non-empty sets:

$A = \{1, 2, 3\}$ $B = \{1, 2\}$

Find the Cartesian Product $A \times B$:

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

Selection – form sub-collections

1. Collect ordered pairs where $x = y$, name it $R_1$:

$R_1 = \{(1,1),\;(2,2)\}$

2. Collect ordered pairs where $x > y$, name it $R_2$:

$R_2 = \{(2,1),\;(3,1),\;(3,2)\}$

3. Collect ordered pairs where $x < y$, name it $R_3$:

$R_3 = \{(1,2)\}$

Observation

$R_1,\; R_2,\; R_3$ are collections of ordered pairs.
$R_1 \subset A \times B$
$R_2 \subset A \times B$
$R_3 \subset A \times B$

Such a collection of ordered pairs $(x, y)$ where $x$ and $y$ have some connection between them, and which is also a subset of $A \times B$, is commonly known as a Relation.

📖 Definition of Relation

Let $A$ and $B$ be any two non-empty sets. A relation $R$ defined from set $A$ to $B$ is a collection of ordered pairs $(x, y)$ from $A \times B$ where $x$ and $y$ have a meaningful full relation.

Mathematically

$$R = \{(x,\,y) : x \in A,\; y \in B\} \subset A \times B$$

Rules used in relations

"Is greater than", "Is less than", "Is equal to", "Is square of" … etc.

⇌ Domain, Range and Co - domain of R

Domain of R

The set of all first elements (antecedents) from the ordered pairs that belong to the relation.

Range of R

The set of all second elements (consequents) from the ordered pairs that belong to the relation.

Example If $R = \{(1,1),\;(2,2),\;(3,3)\}$
Domain of $R = \{1,\;2,\;3\}$

Range of $R = \{1,\;2,\;3\}$

Co - domain of R

If $R$ is a relation defined from set A to B and $R \subset A\times B$, then set B is called the co - domain of relation $R$.

Example:

Inverse Relation

Definition

If $\textbf {R}$ is a relation from set $\textbf {A}$ to set $\textbf {B}$, then the inverse relation of $\textbf {R}$, denoted by $R^{-1}$, is obtained by reversing the ordered pairs.

$$(a,b)\in R$$

then

$$(b,a)\in R^{-1}$$

Mathematical Form

If $$R=\{(a,b)\mid a\in A,\ b\in B\}$$

then

$$R^{-1}=\{(b,a)\mid b\in B, a\in A\}$$

Example 1

Let

$$R=\{(1,2),(2,3),(4,5)\}$$

Then

$$R^{−1}=\{(2,1),(3,2),(5,4)\}$$

🗂 Ways of Representation of a Relation

Setup

Let $A = \{1,2,3\}$, $B = \{1,2\}$. $A \times B = \{(1,1),(1,2),(2,1),(2,2),(3,1),(3,2)\}$.

Define relation $R$ as "Is equal to"

a) As a Set of Ordered Pairs

$R = \{(1,\;1),\;(2,\;2)\}$

b) Arrow or Mapping Diagram

c) Tabular Form

$x$	1	2
$y$	1	2

d) Graphical Form

e) Set Builder Form

$$R = \{(x,\,y) : x = y,\; x \in A,\; y \in B\}$$

🔢 Types of Relations

Context Let $A = \{1, 2, 3\}$. Then: $$A \times A = \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$$

Reflexive Relation

A relation which contains ordered pairs $(x,y)$ such that for all $(x,y) \in R$, either $(x,x) \in R$ or $(y,y) \in R$.

$R = \{(1,1),\;(2,2),\;(3,3)\}$

Every element relates to itself.

Symmetric Relation

A relation which contains ordered pairs $(x,y)$ such that for all $(x,y) \in R$, $(y,x) \in R$.

$R = \{(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\}$

If $a$ relates to $b$, then $b$ relates to $a$.

Transitive Relation

A relation where for all $(x,y) \in R$ and $(y,z) \in R$, we have $(x,z) \in R$.

$R = \{(1,2),\;(2,3),\;(1,3)\}$

If $a \to b$ and $b \to c$, then $a \to c$.

Equivalence Relation

A relation $R$ is said to be an Equivalence Relation if and only if it is simultaneously:

Reflexive
Symmetric
Transitive

$R = \{(1,1),\;(1,2),\;(2,1),\;(2,3),\;(1,3)\}$

New Millennium Academy, Virauta, Pokhara-17 · Mathematics Notes · Grade 9