Class 11 Maths Relation Functions Relation as subset of Cartesian product

Relation: A subset of Cartesian product

A relation R from set A to set B is a subset of the Cartesian product A × B.   The subset is derived by describing a relationship between elements of A & B. 

E.g.: Lets take set A ={a,b,c} & set B ={Amit, Bittu, Bholi , Don}

A*B = { (a,amit),(a,Bittu), (a,Bholi), (a,Don), (b,amit),(b,Bittu), (b,Bholi), (b,Don), (c,amit),(c,Bittu), (c,Bholi), (c,Don)}

Thus A*B has 12 elements

Now if we put a condition (relation), saying first letter of Element in Set B should be the Set A element.

With this condition we get new set as

Conditional Set C = (a,Amit), (b,Bittu), (b, Bholi)}

Set C is sub set of A*B. Thus we say that a relation R from set A to set B is a subset of the Cartesian product A × B

Second element is called image of first element. E.g. “Amit” is image of “a”.

Domain: The set of all first elements of the ordered pairs in a relation.  Eg: a & b not c are domain

 Range: The set of all second elements in a relation R .  E.g. Amit , Bittu and Bholi are range but “Don” is not part of range.

 Codomain: Whole set B . Note that range ⊆ codomain. E.g. : Amit , Bittu Bholi and Don are part of codomain.

 

 

A relation may be represented by :

  • Roster form
  • Set-builder method.
  • An arrow diagram

 

Numerical: Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x, y) : y = x +2 }. Depict this relation using an arrow diagram & set builder form.  Write down the domain, codomain and range of R.

Solution: Lets fist create arrow diagram

Relation A X A = (1,3) ,(2,4) , (3, 5) , (4,6)

Number of Relations

It is number of possible subsets of A × B. 

If n(A ) = p and n(B) = q, then n (A × B) = pq , Thus total number of relations/subset  is 2pq .      Example: Let’s find number of relation for A = {1, 2} and B = {3,4,5}.

Here p=2 & q=3 , so pq = 2*3 = 6

26 = 64, thus it can have 64 relations. Note that relation R from A to A is also stated as a relation on A.

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