One-to-One function is also called injective function [3]
Onto Functions (Surjections)
is onto iff:
Onto function is also called surjective function [4]
One-to-One and Onto Functions (Bijections)
is a bijective function iff
is both onto and one-to-one
One-to-One and onto function is also called bijective function [5]
Relationship between domain, co-domain and range
Injection:
Surjection: , But may not equal to
Bijection:
Image, Pre-image and Range
If from set A to set B, then
is called the image of under
is called a pre-image of
The set of all the images of the elements in the domain under is called the range of .
Inverse Function
is a bijection.
The inverse of is bijection for all x
if then
Composition Function
If is a function from to and is a function from to , then is the composition of g and f
Two special functions
Floor Function
Denoted
Ceiling Funcation
Denoted eg.
Sequences
Sequences are ordered lists of elements
A sequence is a function from a subset of the set of integers or to a set , denoted . The integers determine the positions of the elements in the list.
Summations
Sequences are very useful in summations.
A summation is the value of the sum of the terms of a sequence.
Some Special Summations
A geometric series is a summation of a geometric progression
Geometric progression:
Geometric series:
A harmonic series is the summation of a harmonic progression