**What is Relation?**

The **association of two or more object using some property** is called Relation.

**For Example**

Consider two subjects; Rachel and John.

Its been told that Rachel is “mother of” John.

Here we have associated two subjects Rachel and John with property ” Mother”.

Here the relation (R) is the keyword **” mother of “**.

**Example 02**

James is husband of Pearl

Here the subjects, James & Pearl are associated with relation ” Husband of “.

Hence Relation (R) is **” husband of “**

**Example 03**

10 is greater than 4.

Here digits 10 & 4 are associated with relation ” is greater than”.

Hence, relation (R) is **” is greater than “**

**Relations in Sets**

The concept of Relations is widely used to develop associations with different sets.

**For Example, **

Consider the two sets A & B with following Relations

A = { Tiger, Kangaroo, Panda, Kiwi }

B = { India, Australia, China, New Zealand }

Now suppose we develop relation (R) ” found in ” for the above sets A & B such that;

⟹ Tiger is found in India

⟹ Kangaroo is found in Australia

⟹ Panda is found in China

⟹ Kiwi is found in New Zealand

We know that R = “found in”.

The above association can also be written in following ways;

⟹ Tiger **R** India

⟹ Kangaroo **R** Australia

⟹ Panda **R** China

⟹ Kiwi **R** New Zealand

The above association can also be written in form of ordered pairs;

Note that there are four ordered pair in set R.

The first element of the ordered pair is from set A and second element is from set B.

**Conclusion**The relation between two sets can be easily defined using ordered pairs.

**Representing Relations**

There are **three ways to represent relation between two sets**;

(i) Roster form

(ii) Set Builder form

(iii Arrow Diagram

**Representing Relation using Roster form **

If A & B are two given sets with relation R.

Then the **relation (R)** is **expressed as set of ordered pairs** where first part of the pair belongs to set A and second part belongs to set B.

For Roster form, the key points are;

⟹ Relation (R) is expressed in set of ordered pair

⟹ In ordered pair, **first element belongs to set A** and **second element belongs to set B**.

**Example 01**

Given below is the set A & B.

A = { -2, -1, 0, 1, 2 }

B = { 0, 1, 4, 9, 10 }

A and B is associated with relation; \mathtt{a^{2}} = b

Where; a ϵ A and b ϵ B.

Arrange the relation R in Roster form.

**Solution**

The set A & B follows the given relation; \mathtt{a^{2}} = b.

Finding all the set element step by step;

⟹ When a = -2 then;

\mathtt{b\ =\ ( -2)^{2}}\\ \\ \mathtt{b\ =\ 4}

⟹ When a = -1 then;

\mathtt{b\ =\ ( -1)^{2}}\\ \\ \mathtt{b\ =\ 1}

⟹ When a = 0 then;

\mathtt{b\ =\ (0)^{2}}\\ \\ \mathtt{b\ =\ 0}

⟹ When a = 1 then;

\mathtt{b\ =\ (1)^{2}}\\ \\ \mathtt{b\ =\ 1}

⟹ When a = 2 then;

\mathtt{b\ =\ ( 2 )^{2}}\\ \\ \mathtt{b\ =\ 4}

As per given relation, we have mapped every element of set A & B.

Now describing relation R in Roster form.

Note that in each ordered pair, the first element belong to set A and second element belong to set B.

**Example 02**

Given below are two sets A & B.

A = { 3, 7, 4 }

B = { 2, 9, 5 }

Relation R is defined as ” is greater than ” for set A & B.

Express the relation R in Roster form.

**Solution**

As per the given relation, let’s check the individual elements.

⟹ Compare first element of set A with set B elements.

Number 3 “is greater than” number 2 of set B.

Hence we get (3, 2) ordered pair.

⟹ Compare second element of set A with Set B elements

Number 7 “is greater than” 2 & 5 of set B

So we get (7, 2) & (7, 5) ordered pair.

⟹ Compare third element of set A with set B elements.

Number 4 “is greater than” number 2 of set B

So we get (4, 2) ordered pair.

Combine all the ordered pair in one set.

R = { (3, 2), (7, 2), (7, 5), (4, 2) }

Hence the above R is the solution in Roster form.

**Representing relation in set builder form **

If A & B are the given sets with relation R, then the relation can also be arranged in set builder form.

**In Set builder form** we understand the relation and **express it in the form of single formula**.

What’s the **difference between Roster and Set Builder form** ?

In Roster form we individually write all the ordered pair in the relation.

While in the set builder form, the relation is expressed by single formula.

**Example 01**

Given below is the set A & B with relation R in Roster form.

A = { 3, 6, 9 }

B = { 1, 5, 8 }

Relation (R) = { (3, 5), (3, 8), (6, 8) }

Express the relation in Set Builder form.

**Solution**

If you observe R, you will find that relation R is defined as ” is less than” for set A & B.**Explanation for R**

Checking the individual elements.

⟹ Compare first element of A with all elements of B.

Number 3 “is less than” 5 & 8

(3, 5) & (3, 8) are the ordered pairs

⟹ Compare second element of A with all elements of B.

Number 6 ” is less than” 8

(6, 8) are the ordered pairs

⟹ Compare third element of A with all elements of B

No number in set B which is less than 9.

So no ordered pairs.

Writing all the ordered pairs in Roster form, we get;

R = { (3, 5), (3, 8), (6, 8) }

Hence, **we confirm that R is described as ” is less than”** in the above sets.

Now expressing R in Set Builder form we get;**R = { (a, b) ; a ϵ A, b ϵ B, a is less than b }**

The above set can be read as:

Relation R is a set containing elements a & b such that;

⟹ element a belongs to set A

⟹ element b belongs to set B

⟹ a is less than b

**Example 02**

Given below is the set A and B with following elements;

A = { 2, 5, 8 }

B = { 1/5, 1/8, 1/2 }

Relation R = { (2, 1/2), (5, 1/5), (8, 1/8) }

Express the relation R in set builder form.

**Solution**

On observing the set R we can say that R is described as ” inverse is “.

**Explanation**

⟹Taking Inverse of first element of A.

2 “inverse is” 1/2

So we get the ordered pair (2, 1/2)

⟹ Taking Inverse of second element of A.

5 “inverse is” 1/5

So we get the ordered pair (5, 1/5)

⟹ Taking Inverse of third element of A.

8 “inverse is” 1/8

So we get the ordered pair (8, 1/8)

Combining all the ordered pair we get;

R = { (2, 1/2), (5, 1/5), (8, 1/8) }

Hence its confirmed the** relation R = “inverse is”**

Expressing the relation in set builder form.**R = { (a, b) ; a ϵ A, b ϵ B, b = 1/a }**

**Representing Relation using Arrow Diagram**

If A & B are the given sets, then we express the relation R by;

⟹ drawing first circle containing set A elements

⟹ drawing second circle with set B elements

⟹ joining the elements A & B as per given relation

**Example 01**

Given below is the set A & B with relation set R.

A = { 1, 2, 3, 4, 5 }

B = { 3, 6, 8, 9, 10 }

R = { (1, 8), (2, 6), (3, 9), (4, 10), (5, 6) }

Express the relation R in arrow diagram.

**Solution**

Follow the below steps;

(a)First circle contain set A elements.

(b) Second circle contain set B elements.

(c) Drawn arrows from set A to set B as per given ordered pair of R.