1. Chapter 1
SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs BY: MISS FARAH ADIBAH ADNAN IMK

2. CHAPTER OUTLINE: PART II
1.2 FUNCTIONS
DEFINITION OF FUNCTION
SPECIAL TYPES OF FUNCTION
INVERSE FUNCTION
COMPOSITION OF FUNCTION

3. 1.2.1 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

4. Example 1.1
Find the domain, range and codomain of .

5. 1.2.2 Special Types of Functions:
ONE 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:

6. 2) 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.

7. 3) BIJECTION Have both one to one and onto. Eg:
Let be the function from
with
Is is a bijection?

8. 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.

9. Example 1.2
1) Let be a function from {a,b,c} to {1,2,3} such that Is invertible? What is its inverse? 2) Let be the function from the set of integers such that . Is invertible? What is its inverse?

10. 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

11. Example 1.3
Let be the function from the set {a,b,c} to itself such that Let be the function from the set {a,b,c} to the set {1,2,3} such that What is the composition of and , and what is the composition of and ?

12. Example 1.4
Let and be the function from the set of integers defined by . What is the composition of and , and and ?