Chapter 1

Chapter 1: Function

Relations A relation connects elements in set A (domain) to elements in set B ( codomain) according to the definition of the relation. Relation between two sets can be represented by (a) Arrow diagram (b) Ordered pairs (c) Cartesian graph Example 1

The relation ‘is a factor of’ between set A = { 3, 4} and set B = {4, 6, 8} can be represented as below: (a) Arrow diagram

(b) Ordered pairs {(3, 6), (4, 4), (4, 8)} (c) Cartesian graph

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Form 4: Additional Maths

Chapter 1

Domain, Codomain, Image, Object and Range In the relation between one set and another, the first set is known as the domain and the second set is known as the codomain. Elements in the domain is called objects , whereas elements in the codomain mapped to the objects is called the image. Elements in the codomain not mapped to the objects are not the image. All images in codomain can be written as a set known as range.

Example 2

The diagram above shows the relation rela tion between set A and set B. Find the domain, codomain and range of the relation. Answer

The domain is {a, b} The codomain is {p, q, r, s} The range is {p, q, r}

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Form 4: Additional Maths

Chapter 1

Type of Relation Types of Relation

1

Diagram

One to One Relation

In this relation, each object in the domain has only one image in the codomain.

2

One to Many Relation

In this relation, one or more than one of the objects in the domain has more than one image in the codomain.

3

Many to One Relation

In this relation, there are more than one objects in the domain that have the same image in the codomain.

4

Many to Many Relation

In this relation, one or more than one of the objects in the domain has more than one image in the codomain, and there are more than one object in the domain that have the same image in the codomain.

Function as a special kind of relation •

•

•

A function is a relation in which every element in the domain has a unique image (exactly one) in the codomain. One-to-one relation and many-to-one relation are examples of a special kind of relation which we call function. A function should has the following properties: All objects must have at least 1 image. o It is not necessarily that all elements in the codomain have object. o

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Form 4: Additional Maths

o o

Chapter 1

An object cannot has more than 1 image. More than 1 object is allowed to have the same image. Function is A Special ind of Relation Relat ion

One to One Relation

Many to One Relation

Relation Which is Not Function

The following relations are not function: 1. One One to to Man Many y Rel Relat atio ion n 2. Many Many to Man Many y Rel Relat atio ion n 3. Relation Relation that that has unmapped unmapped object in the the codomain codomain.. 4. For a function function the codomain Y.

, each element x in the domain domain X has a unique unique image y in

5. We often often say y is a functio function n of x and write write it as 6. The function function may also be written as domain to its image f(x) in the *codomain.

. , linking linking an element element of the

7. Note that, the vertical vertical stroke stroke on the arrow distinguis distinguishes hes it from also called the value of the function f at x. 8. f(x) f(x) is is read read as "f of x". x".

. f(x) is

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