2.3 Relations

NCERT Class 11 Mathematics for low vision students.

Consider the two sets


P = {a, b, c} and


Q = {Ali, Bhanu, Binoy, Chandra, Divya}.

The Cartesian product of P and Q has 15 ordered pairs which can be listed as


P × Q =


{(a, Ali), (a,Bhanu), (a, Binoy), ..., (c, Divya)}.

We can now obtain a subset of P × Q by introducing a relation R between the first element x and the second element y of each ordered pair (x, y) as


R = { (x,y): x is the first letter of the name y, x ∈ P, y ∈ Q}.

Then R = {(a, Ali), (b, Bhanu), (b, Binoy), (c, Chandra)}

A visual representation of this relation R (called an arrow diagram) is shown in Figure 2.4.

Figure 2.4

Definition 2

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The second element is called the image of the first element.

Definition 3

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.

Definition 4

The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the co-domain of the relation R.

Note that range ⊂ co-domain.

Remarks:


(i) A relation may be represented algebraically either by the Roster method or by the Set-builder method.


(ii) An arrow diagram is a visual representation of a relation.

Example 7

Let


A = {1, 2, 3, 4, 5, 6}.


Define a relation R from A to A by


R = {(x, y) : y = x + 1 }


(i) Depict this relation using an arrow diagram.


(ii) Write down the domain, co-domain and range of R.

Solution:

(i) By the definition of the relation, we get,


R = {(1,2), (2,3), (3,4), (4,5), (5,6)}.

The corresponding arrow diagram is shown in Figure 2.5.

Figure 2.5

(ii) We can see that the domain = {1, 2, 3, 4, 5}


Similarly, the range = {2, 3, 4, 5, 6} and


the co-domain = {1, 2, 3, 4, 5, 6}.

Example 8

The Figure 2.6 shows a relation between the sets P and Q. Write this relation


(i) in set-builder form,


(ii) in roster form.


What is its domain and range?

Figure 2.6

Solution:

It is obvious that the relation R is “x is the square of y”.

(i) In set-builder form,


R = {(x, y): x is the square of y, x ∈ P, y ∈ Q}

(ii) In roster form,


R = {(9, 3), (9, −3), (4, 2), (4, −2), (25, 5), (25, −5)}

The domain of this relation is {4, 9, 25}.

The range of this relation is {− 2, 2, −3, 3, −5, 5}.

Note that the element 1 is not related to any element in set P.

The set Q is the co-domain of this relation.

Number of Relations

Note that The total number of relations that can be defined from a set A to a set B is the number of possible subsets of the product A × B.


If n(A) = p, and n(B) = q, then n (A × B) = pq,


and the total number of relations is 2^pq.

Example 9

Let A = {1, 2} and B = {3, 4}. Find the number of relations from A to B.

Solution:

We have,


A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.

Since n (A × B ) = 4, the number of subsets of A × B is 2^4. Therefore, the number of relations from A into B will be 2^4 = 16.

Remark: A relation R from A to A is also stated as a relation on A.

EXERCISE 2.2

Question 1.


Let A = {1, 2, 3, ..., 14}.


Define a relation R from A to A by


R = {(x, y) : 3x − y = 0, where x, y ∈ A}.


Write down its domain, co-domain and range.

Answer 1.


R = {(1, 3), (2, 6), (3, 9), (4, 12)}

Domain of R = {1, 2, 3, 4}


Range of R = {3, 6, 9, 12}


Co-domain of R = {1, 2, ..., 14}

Question 2.


Define a relation R on the set N of natural numbers by


R = {(x, y) : y = x + 5, x is a natural number less than 4; x, y ∈ N}.


Depict this relationship using roster form.


Write down the domain and the range.

Answer 2.


R = {(1, 6), (2, 7), (3, 8)}


Domain of R = {1, 2, 3}


Range of R = {6, 7, 8}

Question 3.


Let,


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


B = {4, 6, 9}.


Define a relation R from A to B by


R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}.


Write R in roster form.

Answer 3.


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

Question 4.


The Figure 2.7 shows a relationship between the sets P and Q. Write this relation

Figure 2.7

(i) in set-builder form


(ii) roster form.


What is its domain and range?

Answer 4.


(i) R = {(x, y) : y = x − 2 for x = 5, 6, 7}


(ii) R = {(5,3), (6,4), (7,5)}.


Domain of R = {5, 6, 7},


Range of R = {3, 4, 5}

Question 5.


Let


A = {1, 2, 3, 4, 6}.


Let R be the relation on A defined by {(a, b): a , b ∈ A, b is exactly divisible by a}.


(i) Write R in roster form


(ii) Find the domain of R


(iii) Find the range of R.

Answer 5.


(i) R = {(1, 1), (1,2), (1, 3), (1, 4), (1, 6), (2, 4), (2, 6), (2, 2), (4, 4), (6, 6), (3, 3), (3, 6)}


(ii) Domain of R = {1, 2, 3, 4, 6}


(iii) Range of R = {1, 2, 3, 4, 6}

Question 6.


Determine the domain and range of the relation R defined by


R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.

Answer 6.


Domain of R = {0, 1, 2, 3, 4, 5}


Range of R = {5, 6, 7, 8, 9, 10}

Question 7.


Write the relation R = {(x, x^3) : x is a prime number less than 10} in roster form.

Answer 7.


R = {(2, 8), (3, 27), (5, 125), (7, 343)}

Question 8.


Let


A = {x, y, z} and


B = {1, 2}.


Find the number of relations from A to B.

Answer 8.


Number of relations from A × B = 2^6 = 64

Question 9.


Let R be the relation on Z defined by


R = {(a, b): a, b ∈ Z, a − b is an integer}.


Find the domain and range of R.

Answer 9.


Domain of R = Z


Range of R = Z