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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