Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Properties of Functions: One-to-One, Onto, Bijective Functions, Sequences, Convergence, Study notes of Discrete Structures and Graph Theory

An overview of functions, their properties, and related concepts such as one-to-one, onto, and bijective functions, sequences, and convergence. Topics include function notation, function as relations, composite functions, one-to-one and onto functions, mathematical functions, and summation symbols. The document also discusses recursive function definitions and the complexity of algorithms.

Typology: Study notes

Pre 2010

Uploaded on 08/09/2009

koofers-user-vj2
koofers-user-vj2 🇺🇸

10 documents

1 / 5

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Functions.
A mapping. Usual notation f: A → B
If
Bbb)a(f,Aa
All members of A are mapped to the set B.
A is the domain of the function f, B, its range or
codomain.
Functions as relations.
Composite functions.
f: A→B and then g: B→C is a mapping g*f such that
g*f: A → C.
a
b
c
dx
y
z
m
AB
pf3
pf4
pf5

Partial preview of the text

Download Properties of Functions: One-to-One, Onto, Bijective Functions, Sequences, Convergence and more Study notes Discrete Structures and Graph Theory in PDF only on Docsity!

Functions. A mapping. Usual notation f: A → B If a^ A, f(a)bbB All members of A are mapped to the set B. A is the domain of the function f, B, its range or codomain. Functions as relations. Composite functions. f: A→B and then g: B→C is a mapping gf such that gf: A → C. a b** c d x y z m A B

One-to-0ne, Onto function A function f: A → B is one-to-one if all its members are mapped, and different members of A are mapped to different members in B. Also called injective functions. A function is onto if all elements of B are images of some elements of A. An onto function is also called a surjective function. A function that is both one-to-one and onto are called bijective functions or a one-to-one correspondence. General features of one-to-one and onto functions For an one-to-one f: A → B we cannot have distinct pairs (a,b), (c,b). a b c c m o n p

What about double indexed sums?            n i 1 0 j n 1 ai bj a 1 bn 1 a 2 bn 2 ... anb 0 What about this kind? (a b a b ... a b ) ... ab (ab ab ... ab ) 2 3 2 4 2 q 2 1 3 1 q p i 1 q j i i j 1              What about product forms?   i  1 ai a 1 a 2 a 3 ...an How about (a a a a ) a (aa a a ) (a aa a ) ... n n 1 n 2 n 3 1 2 3 4 2 3 4 5 n i 1 i 3 i i         ^  We can have all kinds of combinations ofand ^ in an expression. Recursive functions definitions. else, gcd(n,m%n ) gcd(m,n) nif(mmodn) 0    Another function.

k k/ 2 2 k k 1 x (x ) else x x x ifk% 2 0      Complexity of algorithms. Big-oh notation,complexity, ^ complexity. Complexities of well-known algorithms. Linear search: O(n) Binary search: O(log n) Bubble sort: O(^ n ) 2 Selection sort: O(^ n ) 2 Insertion sort: O^ (n ) 2 Quicksort: O(n log n)