ICSE Class 8 Maths Chapter 40 Relations and Mappings

Chapter 40: Relations And Mappings

40.1 Introduction

In our daily life, we come across many statements which show some connection between two objects. For example:

(i) Meeta is sister of Ankur, shows connection between two persons

(ii) 7 is greater than 2, shows connection between two numbers

(iii) Line AB is perpendicular to line CD, shows connection between two lines, etc.

Such statements which show some connection (or association or correspondence) between two objects give rise to the concept of relation. A relation means an association of two objects based on some properties possessed by them.

The letter R is generally used to represent a relation. Re-consider the example given above, in which:

(i) the first statement shows a relation between two persons and the relation R = "is sister of".

(ii) the second statement shows a relation between two numbers and the relation R = "is greater than" and so on.

Teacher's Note

Relations help us organize information about how different objects or numbers connect to each other, similar to how we organize contact details in a phonebook.

40.2 Representation Of A Relation

1. Roster Form (as the set of ordered pairs)

For example:

If A = {1, 3, 4, 7, 9, 10, 16}, B = {0, 1, 2, 3, 4, 5} and the relation R from A to B "is square of", then R = { (1, 1), (4, 2), (9, 3), (16, 4) }.

1. Here the relation R is from set A to set B so the first component of each ordered pair is taken from set A and the second component from set B such that, the first component is the square of the second component.

2. If the first component as well as the second component of each ordered pair are taken from set A only, then the relation is called a relation in set A. Similarly, a relation in set B means, the first component as well as the second component of each ordered pair are from set B itself.

3. The set of first components of all the ordered pairs is called the domain and the set of second components is called the range of the relation. Thus, in the example given above, Domain = {1, 4, 9, 16} and Range = {1, 2, 3, 4}

2. Set-Builder Form

Let a relation R from set A to set B means "is greater than" ; then it can be expressed as :

R = { (x, y) : x ∈ A, y ∈ B and x > y }

Therefore, in a set builder form, the relation from set A to set B, is written in the form { (x, y) : x ∈ A, y ∈ B and ________ y }, the blank is to be replaced by the rule which associates x and y.

3. By arrow diagrams

For a relation from set A to set B, arrows are drawn to indicate the pairing, which satisfy the given relation.

1. The arrow heads should indicate the direction from A to B.

2. If the relation R is from set B to set A, the arrow heads should indicate the direction from B to A.

In the example given above, the arrow diagram will be of the form as shown alongside:

In the arrow diagram shown, set A contains {1, 3, 4, 7, 9, 10, 16} and set B contains {0, 1, 2, 3, 4, 5}. Arrows point from elements in A to their related elements in B.

Teacher's Note

Arrow diagrams visually show how items from one group match with items in another, much like showing which students are friends with each other in a class.

Test Yourself

1. Let A = {2, 5, 7, 6}; write the set of all possible ordered pairs satisfying the given relation in set A :

(a) R, = 'is less than' = ____________________________

(b) R, = 'is greater than' = ________________________

(c) R, = 'is equal to' = ______________________________

2. If A = {0, 1, 2, 3, 4, 5, 6, 7} and B = {1, 2, 3, 4, 5, 6}; write the relation R from set A to set B; where R = 'is 2 less than' = ________________

Also write :

(a) the domain of relation R = _______________

(b) the range of relation R = __________________

Represent the relation R using arrow diagram

3. Let A = {7, 8, 9} and B = {5, 6, 7, 8, 9} and a relation R from A to B such that R = {(x, y) : x ∈ A, y ∈ B x ≤ y}; then R = __________________________________

40.3 Mapping Or Function

Mapping or function is a special type of relation. Let A and B be two sets such that A = {a₁, a₂, a₃} and B = {b₁, b₂, b₃, b₄}. If by some rule, each element of set A is associated with a unique element of set B, say a₁ is associated with b₁, a₂ is associated with b₂ and a₃ is associated with b₃, then the collection {(a₁, b₁), (a₂, b₂), (a₃, b₃)} of such associations is called function from A into B.

If this function is denoted by f, then we write :

f : A → B and is read as "f is a function from A to B".

The word 'mapping' is often used as synonym for 'function'.

The set A is called the domain and the set B is called co-domain or range of the function f.

Necessary Conditions For Mapping (Function)

For a function f from set A and set B ; Every element of set A should be associated to a unique element of set B.

i.e. (i) there should not be any element in A which is not associated with any element of B and (ii) no element of A should be associated with two or more elements of B.

1. A function is a special type of relation ; so every function is a relation but the converse is not always true.

2. A relation from A to B, represented in roster form, is a function (mapping) if :

(a) each element of A is associated with unique element of B ;

(b) no two ordered pairs have the same first component i.e. the first components of all the ordered pairs are different.

Example 1

State, giving reason, whether each of the following relations from A to B is a function or not.

(i) If A = {1, 2, 3} and B = {4, 5, 6} ; then R = { (1, 4), (2, 6), (3, 6) }.

(ii) If A = {5, 7, 9} and B = {2, 4} ; then R = { (5, 2), (5, 4), (7, 4), (9, 4) }.

(iii) If A = {a, l, m, n} and B = {x, y, z} ; then R = { (a, x), (l, y), (m, z) }.

Solution

(i) The relation R = { (1, 4), (2, 6), (3, 6) } is a function, as each element in A has been associated with a unique element in B and no two ordered pairs have their first components the same.

(ii) The relation R = { (5, 2), (5, 4), (7, 4), (9, 4) } is not a function, as the element 5 ∈ A has been associated to two elements 2 and 4 in B.

For a relation to be a function, the second component of the ordered pairs may repeat, but the first component cannot repeat.

(iii) The relation R = { (a, x), (l, y), (m, z) } is not a function, as the element n ∈ A is not associated with any element in B.

Arrow Diagram Recognition of Functions

A relation from A to B represented by an arrow diagram, is a function, if :

(a) one and only one arrow connects an element in A to a particular element in B.

(b) there is no element in A, which is not connected (associated) to any element in B.

Example 2

State, giving reason, whether each of the following arrow diagrams represent a function or not :

The document shows six arrow diagrams labeled (i) through (vi). In diagram (i), set A has elements a, b, c connected to set B with elements 1, 2, 3 with each element in A having exactly one arrow to B. In diagram (ii), element a has multiple arrows pointing to different elements in B. In diagram (iii), element c has no arrow. In diagram (iv), element a has multiple arrows. In diagram (v), elements from A have arrows to B in a more complex pattern. In diagram (vi), element c has multiple arrows pointing to different elements.

Solutions

(i) The given arrow diagram represents a function, as each element in A is associated (connected) to a unique element in B.

(ii) It represents a function for the same reason as given in (i).

(iii) The given arrow diagram does not represent a function, as element b ∈ A is connected to two elements in B and also the element c ∈ A is not connected to any element in B.

(iv) It does not represent a function as element a ∈ A is connected to three elements in B.

(v) It represents a function for the same reason as given in (i).

(vi) It does not represent a function, as element c ∈ A is connected to two elements in B.

Teacher's Note

Functions are like vending machines where each input (button pressed) gives exactly one output (item dispensed), ensuring a one-to-one relationship.

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