Relations

Relations

Many day-to-day occurrences involve two objects that are connected with each other by some rule of correspondence. We say that the two objects are related under the specified rule. How shall we represent it? Here are some examples,

How are New  Delhi and  India related? We may  expect the response, “New  Delhi is the capital of India”. But there are several ways in which ‘New Delhi’ and ‘India’ are related. Here are some possible answers.

·           New Delhi is the capital of India. 

·           New Delhi is in the northern part of India.

·           New Delhi is one of the largest cities of India etc.

So, when we wish to specify a particular relation, providing only one ordered pair

(New Delhi, India) it may not be practically helpful. If we ask the relation in the following set of ordered pairs,

{(New Delhi, India), (Washington, USA), (Beijing, China), (London, U.K.), (Kathmandu, Nepal)} then specifying the relation is easy

Illustration 4

Let us define a relation between heights of corresponding students. (Fig.1.7)

R = {(heights, students)}

 R = {(4.5, S₁ ), (4.5, S₄), (4.7, S₉), (4.9, S₁₀), (5, S₃), (5, S₅),(5, S₈), (5.1, S₆),(5.2, S₂),(5.2, S₇)}

Definition

Let A and B be any two non-empty sets. A ‘relation’ R from A to B is a subset of A ×B satisfying some specified conditions. If x ∈ A is related to y ∈ B through R , then we write it as x Ry. x Ry if and only if (x , y) ∈ R .

The domain of the relation R = {x ∈ A | x R y, for some y ∈ B}

The co-domain of the relation R is B

The range of the relation R = {y ∈ B | x Ry, for some x ∈ A}

From these definitions, we note that domain of R ⊆ A , co-domain of R = B and range of R ⊆ B .

Illustration 5

Let A = {1,2,3,4,5} and B = {Mathi, Arul, John}

A relation R between the above sets A and B can be represented by an arrow diagram (Fig. 1.8).

Then, domain of R = {1,2,3,4}

range of R= {Mathi, Arul, John} = co-domain of R . Note that domain of R is a proper subset of A.

Illustration 6

Let A = {1,3,5,7} and B = {4,8}. If R  is a relation defined by “is less than” from A to B, then 1R4 (since 1 is less than 4). Similarly, it is  observed that 1R8, 3R4, 3R8, 5R8, 7R8

Equivalently R= {(1,4), (1,8), (3,4), (3,8), (5,8), (7,8)}

Notes

In the above illustration A×B = {(1,4), (1,8), (3,4), (3,8), (5,4), (5,8), (7,4),(7,8)}

R = {(1,4), (1,8), (3,4), (3,8), (5,8), (7,8)} We see that R is a subset of A×B

Illustration 7

In a particular area of a town, let us consider ten families A, B, C, D, E, F, G, H, I and J with two children. Among these, families B, F, I have two girls; D, G, J have one boy and one girl; the remaining have two boys. Let us define a relation R by xRy, where x denote the number of boys and y denote  the family with x number of boys. Represent this situation as a relation through ordered pairs and arrow diagram.

Since the domain of the relation R is concerned about the number of boys, and we are considering families with twochildren, the domain of R will consist of three elements given by {0,1,2}, where 0, 1, 2 represent the number of boys say no, one, two boys respectively. We note that families with two girls are the ones with no boys. Hence the relation R is given by

R = {(0, B),(0, F),(0, I ),(1, D),(1,G),(1,J ),(2, A),(2,C ),(2, E),(2, H )}

This relation is shown in an arrow diagram (Fig.1.9).

Example 1.4

Let A = {3,4,7,8} and B = {1,7,10}. Which of the following sets are relations from A to B?

(i) R₁ ={(3,7), (4,7), (7,10), (8,1)}

(ii) R₂= {(3,1), (4,12)}

(iii) R₃= {(3,7), (4,10), (7,7), (7,8), (8,11), (8,7), (8,10)}

Solution   A×B = {(3,1), (3,7), (3,10), (4,1), (4,7), (4,10), (7,1),  (7,7), (7,10), (8,1), (8,7), (8,10)}

(i) We note that, R₁ ⊆ A× B . Thus, R₁ is a relation from A to B.

(ii) Here, (4,12) ∈ R₂ , but (4,12) ∉ A×B . So, R₂ is not a relation from A to B.

(iii) Here, (7, 8) ∈ R₃ , but (7, 8) ∉ A×B . So, R₃ is not a relation from A to B.

Notes

A relation may be represented algebraically either by the roster method or by the set builder method.

An arrow diagram is a visual representation of a relation.

Example 1.5 The arrow diagram shows (Fig.1.10) a relationship between the sets P and Q. Write the relation in (i) Set builder form (ii) Roster form (iii) What is the domain andrange of .

Solution

(i) Set builder form of R  = {(x,y)| y = x − 2, x ∈ P,y ∈ Q}

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

(iii) Domain of R = {5,6,7} and range of R = {3, 4, 5}

‘Null relation’

Let us consider the following examples. Suppose A = {–3,–2,–1} and B = {1,2,3,4}. A relation from A to B is defined as a − b = 8 i.e., there is no pair (a,b) such that a − b = 8 . Thus R contain no element and so R = ɸ .

A relation which contains no element is called a “Null relation”.