After deriving functions using
physics, it was used to represent all mathematical operations in a generalized,
common and a simple way. Differentiation and integration were created
only after these functions.
It has to be
represented everything in a simplest way on dealing with physics and
mathematics.
As it have been studied in the
functions, it was represented in the generalized version as y=f(x).
[Note: Symbols are not important.
Don’t focus on the symbols and be not confused with them. e.g. y=f(x) is same
as a=b(X)]
Depending on the situation “x
the input" can have many values. And those values are symbolized with the
Number system.
E.g. In speed(s) = distance
(d) / time (t)
The distance (d) which is an input
can have values from “zero” to “infinity”. But it cannot have negative values
as the distance is always positive. Mathematically the distance (d) can be from zero to infinity.
We represent all these values in a
single Notation called “Sets”.
Sets are
the representation of a group of all possible values of one variable (“x” or “y”)
in a function.
It was denoted using two brackets “{ }” and the “possible values” written
inside the brackets. This notation is called the Set builder Notation.
The possible values are called the “elements” of a set.
E.g. Set of all possible
values of distance (d) is denoted by “D = {x | x is from zero to infinity}”
where the possible values of x is
called the elements of the set D.
In mathematics we have already seen that the set of Whole numbers were denoted
by,
“W= {0,1,2,3,4,….. up to the
infinity}” where 0,1,2,3,4,…etc. are the elements of the set of W.
Set of integers were denoted by,
“Z={negative
infinity,…..-1000,…-100,…-4,-3,-2,-1,0,1,2,3,4,….positive infinity}”
Set of Real numbers were denoted by,
“R={negative
infinity,…..-0.11111,…-0.1,..-0.00001,…0,…0.1,..0.2345..,…positive infinity}”
It contains every
possible real values which includes rational and irrational numbers.
Also it can be seen that the “set D
= {x | x is from zero to infinity}” is the same set that is, “the set of non-
negative real numbers which includes the zero”.
Note: As the knowledge of Number system
made us easy to understand the sets,Now it can be understood why the set of
Natural Numbers, Whole Numbers and Real Numbers were already defined in our
elementary mathematics.
Thus the function is y=f(x), and Y={set of all possible values of y} and
X= {set of all possible values of x}.
We can also say that “the function
is from X to Y” because for each element in X there is an output element in Y.
It means that the machines starts from the input and finally gives the result
of output.
Mathematically, the set X which is
the set of inputs is called the Domain of the function f( ).
But what is set Y? How can it be
determined the all possible values of “y” in a function?
What if “the number of elements in the set
Y is greater than the number of elements in the set X?”
Is it possible to have a few
elements in the set Y are present which are not the outputs of any element in
set X? Does it have any confusion with the definition of the function?
Excess number of elements in the set
Y that is not the outputs of the input set X doesn’t cause any confusion when
we are dealing it with physics and mathematics problems as it always cannot be
sure of the output values.
Hence the set Y can have excess
values than the outputs f(x). It is like excess wastages coming out of the
machine which is not the required output.
Therefore the description of set Y becomes as “the set of all possible outputs
and the excess elements other than outputs”.
The set Y was named a new name called the Codomain of the function.
The elements of the set Y are called the images of the input values of the set
X.
Each “y” value is called the image of “x” under the function “f” and each “x”
is called the pre-image of “y”.
And the set of all possible
“outputs” or “images” are called the Range of the function.
Sometimes it may happen that the set
Y has less number of elements than the set X.
It is also possible to get the same
output value for different input values. E.g. Constant function is the function
where all the input values of the domain have unique output value that is a
constant.
If the Range and Codomain is same in a function, then the
function is called Onto function or Surjective function and if Codomain
have excess elements than the Range then the function is Into function.
Thus the conclusion of the definition is,
“A function from X to Y is a rule or correspondence that assigns to
each element of set X, one and only one element of set Y”. Let the rule or correspondence
be ‘f( )’, then the function from X to Y can be written mathematically as f:X→Y [where “→”denotes the meaning “to”]
where y=f(x), and x € X and y € Y [“€” denotes that “subset of” or “the element
of”]
and ‘y’ is the image of ‘x’ under ‘f( )’.
X is the domain of the function,
Y is the codomain of the function,
and the set of output elements is the Range of the function.
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