physicscatalyst.com logo




Relation in Maths | Definition| Pictorial Representation





2. What is relation?

Definition: A relation \(R\) from a non-empty set \(A\) to a non-empty set \(B\) is a subset of the Cartesian product \(A \times B\).
It "maps" elements of one set to another set. The subset is derived by describing a relationship between the first element and the second element of the ordered pair \(\left( {A \times B} \right)\).
Domain: The set of all first elements of the ordered pairs in a relation \(R\) from a set \(A\) to a set \(B\) is called the domain of the relation \(R\).
Range: the set of all the ending points is called the range
Co-domain: The whole set B is called the co-domain of the relation R.
Example
Let A ={4,5,3} and B ={1,6, 7}
Let R be the Relation "is greater than " from A to B
Then
R= { (4,1),(5,1),(3,1)}

Algebraic Representation of Relation

A relation can be expressed in Set builder or Roaster form

Roster forms

In a Roster forms, all the ordered pair in the relation is listed.
Example
R= { (4,1),(5,1),(3,1)}

Some Important points

  • In roster form, the order in which the elements are listed is immaterial
  • while writing the set in roster form an element is not generally repeated

Set Builder Form

  • In set-builder form, all the ordered pair of a relation possess a single common property which is not possessed by any ordered pair outside the relation. For example, in the relation \(\left\{ (1,2) ,(2,4) ,(3,6) ,(4,8),(5,10) \right\}\), all the ordered pair possess a common property, namely, second element in ordered is doubled of first element . Denoting this set by \(R\), we write
    $R = \left\{(x,y) : x,y \in {1,2,3,4,5,6,7,8,9,10},y=2x \right\}$

Pictorial Representation of Relation

A Relation R from A to B can be depicted pictorially using arrow diagram . In arrow diagram, we write down the elements of two set A and B in two disjoint circle,Then we draw arrow from set A to set B whenever $(a,b) \in R$

Let A={a,b,c,d} and B={x,y,z}
And R ={(a,x),(b,y),(c,z)}
Then this will be represented in arrow diagram as
Arrow diagram for Relations in Maths

Important Note

The total number of relations that can be defined from a set \(A\) to a set \(B\) is the number of possible subsets of \(A \cdot B\). If \(n\left( A \right) = p\) and \(n\left( B \right) = q\), then \(n\left( {A \cdot B} \right) = pq\) and the total number of relations is \({2^{pq}}\)
Example:
Let \(P = \left\{ {1,2,3,.....,18} \right\}\) define a relation \(R\) from \(P\) to \(P\) by \(R = \left\{ {\left( {x,y} \right):2x - y = 0,where \; x,y \in P} \right\}\) Write down its domain, co-domain and range.
Draw the arrow diagram for the relation also
Solution: The relation \(R\) from \(P\) to \(P\) is given as
R = {(x,y):2x-y=0, where x, y ∈ P}
i.e., R = {(x, y): 2x = y, where x, y ∈ P}
Therefore,
\( R = \left\{ {\left( {1,2} \right),\left( {2,4} \right),\left( {3,6} \right),\left( {4,8} \right),\left( {5,10} \right),\left( {6,12} \right),\left( {7,14} \right),\left( {8,16} \right),\left( {9,18} \right)} \right\}\)
The domain of \(R\) is the set of all first elements of the ordered pairs in the relation.
Therefore,
\(Domain \; of \; R = \left\{ {1,2,3,4,5,6,7,8,9} \right\}\)
The whole set \(P\) is the co-domain of the relation \(R\).
Therefore co-domain of \(R = P = \left\{ {1,2,3, \ldots ,18} \right\}\)
The range of \(R\) is the set of all second elements of the ordered pairs in the relation.
Therefore range of \(R = \left\{ {2,4,6,8,10,12,14,16,18} \right\}\)
Arrow diagram is given below
arrow diagram of relations in maths



Quiz Time

Question 1 Let P ={a,b,c} and Q={x,y,z} then which is of the below is not a relation from P to Q
A. R = {(a,x),(b,y),(c,z)}
B. R = {(a,y),(b,z), (c,x)}
C. R = {(x,a),(b,y),(c,z)}
D. None of the above
Question 2Let A = {1, 2, 3,4} and B = {5, 7}. Then possible number of relation from A to B ? A. 64
B. 256
C. 16
D. 32
Question 3 A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by $R = \left \{ (x, y): the \; difference \; between \; x \; and \; y \; is \; odd; x \in A, y \in B \right \}$. R is roster form is given by
A. $R ={(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)}$
B. $R ={(1,4),(1,6),(2,9),(3,4),(3,6),(5,6)}$
C. $R ={(1,4),(1,6),(2,9),(3,4),(3,6)}$
D. $R ={(1,4),(1,6),(2,9),(3,4),(3,6),(4,5),(5,6)}$
Question 4 Let A = {1, 2, 3,4, 5,6,7,8,9} and R is relation defined on Set A as R ={(1,1),(2,4),(3,9)}. What is the range of the relation?
A. {1,2,3,4,5,6,7,8,9}
B. {1,2,4,3,9}
C. {1,4,9}
D. {1.2.3}
Question 5 What is the domain in the above relation
A. {1,2,3,4,5,6,7,8,9}
B. {1,2,4,3,9}
C. {1,4,9}
D. {1.2.3}
Question 6Check the below arrow diagram and find out the relation in roster form

A. R={(7,4),(8,5),(9,6)}
B. R={(4,7),(5,8),(6,9)}
C. R={(7,3),(8,3),(9,3)}
D. R={(3,4),(3,5),(3,6)}



link to this page by copying the following text

Note to our visitors :-

Thanks for visiting our website. From feedback of our visitors we came to know that sometimes you are not able to see the answers given under "Answers" tab below questions. This might happen sometimes as we use javascript there. So you can view answers where they are available by reloding the page and letting it reload properly by waiting few more seconds before clicking the button.
We really do hope that this resolve the issue. If you still hare facing problems then feel free to contact us using feedback button or contact us directly by sending is an email at [email protected]
We are aware that our users want answers to all the questions in the website. Since ours is more or less a one man army we are working towards providing answers to questions available at our website.


Class 11 Maths Class 11 Physics