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.