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 pairs 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
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
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