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Function

Definitions on function

A function is something that associates each element of a set with an element of another set (which may or may not be the same as the first set). The concept of function appears quite often even in non-technical contexts. For example, a social security number uniquely identifies the person, the income tax rate varies depending on the income, and the final letter grade for a course is often determined by test and exam scores, homeworks and projects, and so on.

In all these cases to each member of a set (social security number, income, tuple of test and exam scores, homeworks and projects) some member of another set (person, tax rate, letter grade, respectively) is assigned.

As you might have noticed, a function is quite like a relation. In fact, formally, we define a function as a special type of binary relation.

Definition (function): A function, denote it by f, from a set A to a set B is a relation from A to B that satisfies

1. for each element a in A, there is an element b in B such that<a, b>is in the relation, and

2. if<a, b>and<a, c>are in the relation, then b = c .

The set A in the above definition is called the domain of the function and B its codomain.

Thus, f is a function if it covers the domain (maps every element of the domain) and it is single valued.

The relation given by f between a and b represented by the ordered pair  <a, b> is denoted as  f(a) = b , and b is called the image of a under f .

The set of images of the elements of a set S under a function f is called the image of the set S under f, and is denoted by  f(S) , that is,

f(S) = { f(a) | a ∈ S }, where S is a subset of the domain A of  f .

The image of the domain under f is called the range of f.

Example: Let f be the function from the set of natural numbers N to N that maps each natural number x to x2. Then the domain and co-domain of this f are N, the image of, say 3, under this function is 9, and its range is the set of squares, i.e. { 0, 1, 4, 9, 16, ....} .

Definition (sum and product): Let f and g be functions from a set A to the set of real numbers R.

Then the sum and the product of f and g are defined as follows:

For all x, ( f + g )(x) = f(x) + g(x) , and

for all x, ( f*g )(x) = f(x)*g(x) ,

where f(x)*g(x) is the product of two real numbers f(x) and g(x).

Example: Let f(x) = 3x + 1 and g(x) = x2 . Then ( f + g )(x) = x2 + 3x + 1 , and ( f*g )(x) = 3x3 + x2

Definition (one-to-one): A function f is said to be one-to-one (injective) , if and only if whenever f(x) = f(y) , x = y .

Example: The function f(x) = x2 from the set of natural numbers N to N is a one-to-one function. Note that f(x) = x2 is not one-to-one if it is from the set of integers (negative as well as non-negative) to N, because for example f(1) = f(-1) = 1 .

Definition (onto): A function f from a set A to a set B is said to be onto(surjective) , if and only if for every element y of B , there is an element x in A such that  f(x) = y ,  that is,  f is onto if and only if  f( A ) = B .

Example: The function f(x) = 2x from the set of natural numbers N to the set of non-negative even numbers E is an onto function. However, f(x) = 2x from the set of natural numbers N to N is not onto, because, for example, nothing in N can be mapped to 3 by this function.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
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what is the stm
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is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
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what is a peer
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is differents between GO and RGO?
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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analytical skills graphene is prepared to kill any type viruses .
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what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Is there any normative that regulates the use of silver nanoparticles?
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what king of growth are you checking .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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why we need to study biomolecules, molecular biology in nanotechnology?
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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why?
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biomolecules are e building blocks of every organics and inorganic materials.
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anyone know any internet site where one can find nanotechnology papers?
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
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