chapter 1. 1.2 functions 1.2.1 definition of function 1.2.2 special types of function 1.2.3 inverse...

SETS, FUNCTIONs, ELEMENTARY SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAsLOGIC & BOOLEAN ALGEBRAs

BY: MISS FARAH ADIBAH ADNANBY: MISS FARAH ADIBAH ADNANIMK IMK

Chapter 1Chapter 1

CHAPTER OUTLINE: PART IICHAPTER OUTLINE: PART II

1.2 FUNCTIONS

1.2.1 DEFINITION OF FUNCTION

1.2.2 SPECIAL TYPES OF FUNCTION

1.2.3 INVERSE FUNCTION

1.2.4 COMPOSITION OF FUNCTION

1.2 Function 1.2 Function 1.2.11.2.1 Definition of Function: Definition of Function: Let and be sets. A function from to , we

write as , is an assignment of all elements in set to exactly one element of .

Symbols for the function, . Sometimes write as

Set is called domain, and set is called range / image.

Image is often a subset of a larger set, called codomain.

X Y

X

X Y

X Y X Y:f X Y

X Yf

x y

X Y

X Y

y f x

Example 1.1Example 1.1

Find the domain, range and codomain of .f

1.2.2 Special Types of Functions: 1.2.2 Special Types of Functions: 1)1)ONE TO ONE / INJECTIVEONE TO ONE / INJECTIVE• A function is said one to one, if and only if • Have a distinct images, at a distinct elements of their

domain.• Eg:

f x f yf

2) ONTO / SURJECTIVE2) ONTO / SURJECTIVE• Let a function from A to B, it is called onto if and

only if for every element , there is an element

.

• Eg: refer textbook.

f

b Ba A

, y Y x X

3) BIJECTION3) BIJECTION• Have both one to one and onto.• Eg:

Let be the function from with

Is is a bijection?

f , , , to 1,2,3,4a b c d

4, 2, 1 and 3.f a f b f c f d

f

1.2.3 Inverse Functions: 1.2.3 Inverse Functions: • Let be a function whose domain is the set , and

the codomain is the set . Then the inverse function, has domain of the set Y and codomain of the set

X, with the property:

• The inverse function exists if and only if is a bijection.

f XY

1f

1 if and only if ff x y y x

f

ExampleExample 1.2 1.2

1) Let be a function from {a,b,c} to {1,2,3} such thatIs invertible? What is its inverse?

2) Let be the function from the set of integers such that . Is invertible? What is its inverse?

f

( ) 2, ( ) 3, and ( ) 1.f a f b f c f

f

( ) 1f x x f

1.2.4 Composition of Functions: 1.2.4 Composition of Functions: • Let be a function from the set A to the set B,

and let be a function from the set B to the set C. The composition of the functions and

, denoted by , is defined by:

• The composition of cannot be defined unless the range of is a subset of the domain

gf

f

g f g

( )( ) ( ( ))f g a f g a

f gg

f

ExampleExample 1.3 1.3

Let be the function from the set {a,b,c} to itself such thatLet be the function from the set {a,b,c} to the set {1,2,3} such thatWhat is the composition of and , and what is the composition of and ?

g( ) , ( ) , and ( ) .g a b g b c g c a

f

( ) 3, ( ) 2, and ( ) 1.f a f b f c

f gg f

Let and be the function from the set of integers defined by .What is the composition of and , and and ?

ExampleExample 1.4 1.4

gf

( ) 2 3 and g(x)=3x+2f x x f g g f