1. A function maps elements from its domain to elements in its codomain, assigning each domain element to exactly one codomain element. Functions can be represented graphically.
2. Key properties of functions include being one-to-one (injective), onto (surjective), and bijective (both one-to-one and onto). The identity function maps each element to itself.
3. For bijective functions, the inverse function maps each element in the codomain back to a unique element in the domain.
(a) Natural Numbers : N = {1,2,3,4,...}
(b) Whole Numbers : W = {0,1,2,3,4, }
(c) Integer Numbers :
or Z = {...–3,–2,–1, 0,1,2,3, },
Z+ = {1,2,3,....}, Z– = {–1,–2,–3, }
Z0 = {± 1, ± 2, ± 3, }
(d) Rational Numbers :
p
Q = { q ; p, q z, q 0 }
(i) R0 : all real numbers except 0 (Zero).
(j) Imaginary Numbers : C = {i,, }
(k) Prime Numbers :
These are the natural numbers greater than 1 which is divisible by 1 and itself only, called prime numbers.
Ex. 2,3,5,7,11,13,17,19,23,29,31,37,41,...
(l) Even Numbers : E = {0,2,4,6, }
(m) Odd Numbers : O = {1,3,5,7, }
Ex. {1,
Note :
5
, –10, 105,
3
22 20
7 , 3
, 0 ....}
The set of the numbers between any two real numbers is called interval.
(a) Close Interval :
(i) In rational numbers the digits are repeated after decimal.
(ii) 0 (zero) is a rational number.
(e) Irrational numbers: The numbers which are not rational or which can not be written in the form of p/q ,called irrational numbers
Ex. { , ,21/3, 51/4, ,e, }
Note:
(i) In irrational numbers, digits are not repeated after decimal.
(ii) and e are called special irrational quantities.
(iii) is neither a rational number nor a irrational number.
(f) Real Numbers : {x, where x is rational and irrational number}
20
[a,b] = { x, a x b }
(b) Open Interval:
(a, b) or ]a, b[ = { x, a < x < b }
(c) Semi open or semi close interval:
[a,b[ or [a,b) = {x; a x < b}
]a,b] or (a,b] = {x ; a < x b}
Let A and B be two given sets and if each element a A is associated with a unique element b B under a rule f , then this relation is called function.
Here b, is called the image of a and a is called the pre- image of b under f.
Note :
(i) Every element of A should be associated with
Ex. R = { 1,1000, 20/6, ,
, –10, –
,.....}
3
B but vice-versa is not essential.
(g) Positive Real Numbers: R+ = (0,)
(h) Negative Real Numbers : R– = (– ,0)
(ii) Every element of A should be associated with a unique (one and only one) element of but
any element of B can have two or more rela- tions in A.
3.1 Representation of Function :
It can be done by three methods :
(a) By Mapping
(b) By Algebraic Method
(c) In the form of Ordered pairs
(A) Mapping :
It shows the graphical aspect of the relation of the elements of A with the elements of B .
Ex. f1:
f2 :
f3 :
f4 :
In the above given mappings rule f1 and f2
shows a function because each element of A is
associated with a unique element of B. Whereas
f3 and f4 are not function because in f 3, element c is associated with two elements of B, and in f4 , b is not associated with any element
of B, which do not follow the definition of function. In f2, c and d are associated with same element, still it obeys the rule of definition of function because it does not tell that every element of A should be associated with different elements of B.
(B) Algebraic Method :
It shows the relation between the elem
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
(a) Natural Numbers : N = {1,2,3,4,...}
(b) Whole Numbers : W = {0,1,2,3,4, }
(c) Integer Numbers :
or Z = {...–3,–2,–1, 0,1,2,3, },
Z+ = {1,2,3,....}, Z– = {–1,–2,–3, }
Z0 = {± 1, ± 2, ± 3, }
(d) Rational Numbers :
p
Q = { q ; p, q z, q 0 }
(i) R0 : all real numbers except 0 (Zero).
(j) Imaginary Numbers : C = {i,, }
(k) Prime Numbers :
These are the natural numbers greater than 1 which is divisible by 1 and itself only, called prime numbers.
Ex. 2,3,5,7,11,13,17,19,23,29,31,37,41,...
(l) Even Numbers : E = {0,2,4,6, }
(m) Odd Numbers : O = {1,3,5,7, }
Ex. {1,
Note :
5
, –10, 105,
3
22 20
7 , 3
, 0 ....}
The set of the numbers between any two real numbers is called interval.
(a) Close Interval :
(i) In rational numbers the digits are repeated after decimal.
(ii) 0 (zero) is a rational number.
(e) Irrational numbers: The numbers which are not rational or which can not be written in the form of p/q ,called irrational numbers
Ex. { , ,21/3, 51/4, ,e, }
Note:
(i) In irrational numbers, digits are not repeated after decimal.
(ii) and e are called special irrational quantities.
(iii) is neither a rational number nor a irrational number.
(f) Real Numbers : {x, where x is rational and irrational number}
20
[a,b] = { x, a x b }
(b) Open Interval:
(a, b) or ]a, b[ = { x, a < x < b }
(c) Semi open or semi close interval:
[a,b[ or [a,b) = {x; a x < b}
]a,b] or (a,b] = {x ; a < x b}
Let A and B be two given sets and if each element a A is associated with a unique element b B under a rule f , then this relation is called function.
Here b, is called the image of a and a is called the pre- image of b under f.
Note :
(i) Every element of A should be associated with
Ex. R = { 1,1000, 20/6, ,
, –10, –
,.....}
3
B but vice-versa is not essential.
(g) Positive Real Numbers: R+ = (0,)
(h) Negative Real Numbers : R– = (– ,0)
(ii) Every element of A should be associated with a unique (one and only one) element of but
any element of B can have two or more rela- tions in A.
3.1 Representation of Function :
It can be done by three methods :
(a) By Mapping
(b) By Algebraic Method
(c) In the form of Ordered pairs
(A) Mapping :
It shows the graphical aspect of the relation of the elements of A with the elements of B .
Ex. f1:
f2 :
f3 :
f4 :
In the above given mappings rule f1 and f2
shows a function because each element of A is
associated with a unique element of B. Whereas
f3 and f4 are not function because in f 3, element c is associated with two elements of B, and in f4 , b is not associated with any element
of B, which do not follow the definition of function. In f2, c and d are associated with same element, still it obeys the rule of definition of function because it does not tell that every element of A should be associated with different elements of B.
(B) Algebraic Method :
It shows the relation between the elem
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
2. 2
Definition of Functions
• Given any sets A, B, a function f from (or
“mapping”) A to B (f:AB) is an
assignment of exactly one element f(x)B
to each element xA.
3. 3
Graphical Representations
• Functions can be represented graphically in
several ways:
• •
A
B
a b
f
f
•
•
•
•
•
•
•
•
• x
y
Plot
Graph
Like Venn diagrams
A B
4. 4
Definition of Functions (cont’d)
• Formally: given f:AB
“x is a function” : (x,y: x=y f(x) f(y)) or
“x is a function” : ( x,y: (x=y) (f(x) f(y))) or
“x is a function” : ( x,y: (x=y) (f(x) = f(y))) or
“x is a function” : ( x,y: (x=y) (f(x) = f(y))) or
“x is a function” : ( x,y: (f(x) f(y)) (x y))
5. 5
Some Function Terminology
• If f:AB, and f(a)=b (where aA & bB),
then:
– A is the domain of f.
– B is the codomain of f.
– b is the image of a under f.
– a is a pre-image of b under f.
• In general, b may have more than one pre-image.
– The range RB of f is {b | a f(a)=b }.
6. 6
Range vs. Codomain - Example
• Suppose that: “f is a function mapping
students in this class to the set of grades
{A,B,C,D,E}.”
• At this point, you know f’s codomain is:
__________, and its range is ________.
• Suppose the grades turn out all As and Bs.
• Then the range of f is _________, but its
codomain is __________________.
{A,B,C,D,E} unknown!
{A,B}
still {A,B,C,D,E}!
7. 7
Function Addition/Multiplication
• We can add and multiply functions
f,g:RR:
– (f g):RR, where (f g)(x) = f(x) g(x)
– (f × g):RR, where (f × g)(x) = f(x) × g(x)
8. 8
Function Composition
• For functions g:AB and f:BC, there is a
special operator called compose (“○”).
– It composes (i.e., creates) a new function out of f,g by
applying f to the result of g.
(f○g):AC, where (f○g)(a) = f(g(a)).
– Note g(a)B, so f(g(a)) is defined and C.
– The range of g must be a subset of f’s domain!!
– Note that ○ (like Cartesian , but unlike +,,) is non-
commuting. (In general, f○g g○f.)
10. 10
Images of Sets under Functions
• Given f:AB, and SA,
• The image of S under f is simply the set of
all images (under f) of the elements of S.
f(S) : {f(s) | sS}
: {b | sS: f(s)=b}.
• Note the range of f can be defined as simply
the image (under f) of f’s domain!
11. 11
One-to-One Functions
• A function is one-to-one (1-1), or injective,
or an injection, iff every element of its
range has only one pre-image.
• Only one element of the domain is mapped
to any given one element of the range.
– Domain & range have same cardinality. What
about codomain?
12. 12
One-to-One Functions (cont’d)
• Formally: given f:AB
“x is injective” : (x,y: xy f(x)f(y)) or
“x is injective” : ( x,y: (xy) (f(x)f(y))) or
“x is injective” : ( x,y: (xy) (f(x) f(y))) or
“x is injective” : ( x,y: (xy) (f(x) f(y))) or
“x is injective” : ( x,y: (f(x)=f(y)) (x =y))
13. 13
One-to-One Illustration
• Graph representations of functions that are
(or not) one-to-one:
•
•
•
•
•
•
•
•
•
One-to-one
•
•
•
•
•
•
•
•
•
Not one-to-one
•
•
•
•
•
•
•
•
•
Not even a
function!
14. 14
Sufficient Conditions for 1-1ness
• Definitions (for functions f over numbers):
– f is strictly (or monotonically) increasing iff
x>y f(x)>f(y) for all x,y in domain;
– f is strictly (or monotonically) decreasing iff
x>y f(x)<f(y) for all x,y in domain;
• If f is either strictly increasing or strictly
decreasing, then f is one-to-one.
– e.g. f(x)=x3
15. 15
Onto (Surjective) Functions
• A function f:AB is onto or surjective or a
surjection iff its range is equal to its
codomain (bB, aA: f(a)=b).
• An onto function maps the set A onto (over,
covering) the entirety of the set B, not just
over a piece of it.
– e.g., for domain & codomain R, x3 is onto,
whereas x2 isn’t. (Why not?)
16. 16
Illustration of Onto
• Some functions that are or are not onto their
codomains:
Onto
(but not 1-1)
•
•
•
•
•
•
•
•
•
Not Onto
(or 1-1)
•
•
•
•
•
•
•
•
•
Both 1-1
and onto
•
•
•
•
•
•
•
•
1-1 but
not onto
•
•
•
•
•
•
•
•
•
17. 17
Bijections
• A function f is a one-to-one
correspondence, or a bijection, or
reversible, or invertible, iff it is both one-
to-one and onto.
18. 18
Inverse of a Function
• For bijections f:AB, there exists an
inverse of f, written f 1:BA, which is the
unique function such that:
I
f
f
1
20. 20
The Identity Function
• For any domain A, the identity function
I:AA (variously written, IA, 1, 1A) is the
unique function such that aA: I(a)=a.
• Some identity functions you’ve seen:
– ing with T, ing with F, ing with , ing
with U.
• Note that the identity function is both one-
to-one and onto (bijective).
21. 21
• The identity function:
Identity Function Illustrations
•
•
•
•
•
•
•
•
•
Domain and range x
y
22. 22
Graphs of Functions
• We can represent a function f:AB as a set
of ordered pairs {(a,f(a)) | aA}.
• Note that a, there is only one pair (a, f(a)).
• For functions over numbers, we can
represent an ordered pair (x,y) as a point on
a plane. A function is then drawn as a curve
(set of points) with only one y for each x.
24. 24
A Couple of Key Functions
• In discrete math, we frequently use the
following functions over real numbers:
– x (“floor of x”) is the largest integer x.
– x (“ceiling of x”) is the smallest integer x.
25. 25
Visualizing Floor & Ceiling
• Real numbers “fall to their floor” or “rise to
their ceiling.”
• Note that if xZ,
x x &
x x
• Note that if xZ,
x = x = x.
0
1
1
2
3
2
3
.
.
.
.
.
.
. . .
1.6
1.6=2
1.4= 2
1.4
1.4= 1
1.6=1
3
3=3= 3
26. 26
Plots with floor/ceiling: Example
• Plot of graph of function f(x) = x/3:
x
f(x)
Set of points (x, f(x))
+3
2
+2
3