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Functions
As used in ordinary language, the word function indicates
dependence of a varying quantity on
another. If I tell you that your grade in this class is a function of your
overall average, you
interpret this to mean that I have a rule for translating a number in the range
of 0 to 100 into a
letter grade. More generally, suppose two sets of objects are given: set A and
set B; and
suppose that with each element of A there is associated a particular element of
B. These three
things: the two sets and the correspondence between elements comprise a
function.
A function f is a mapping from a set D to
a set T with the property that for each element d in D, f maps d to a single element of T, denoted f(d). Here D is called the domain of f, and T is called the target or codomain. We write f: D → T. We also say that f(d) is the image of d under f, and we call the set of all images the range R of f. 
A mapping might fail to be a function if it is not defined
at every element of the domain, or if
it maps an element of the domain to two or more elements in the range:
Consider the mappings shown in the diagram above. We note
that examples a and b are both
functions since every element in the first set (the domain) maps to a single
element in the second
set (the codomain or target). Note that it is fine for two elements in the
domain to map to the
same element in the codomain (as is the case in b). We point out that
example c is not a function,
since there is an element in the domain which does not map to any element in the
codomain.
Also, example d is not a function since there is an element in the domain
that maps to more than
one element in the codomain.
One way to define a function is to provide a table that shows the mapping for
each element of the
domain. For example, in a small class we might have
the set of students S = {Maria, Clara. Tom, Dick, Harry}
the set of possible grades G = {A, B, C, D, NP}
One possible function f: S → G would be:
d  f(d) 
Maria Clara Tom Dick Harry 
A B C A B 
The table completely defines the function by showing every
mapping. Note that all the
possible grades are not used, i.e., the range is not the same as the codomain.
It is still
convenient to call this a function from S to G in order to indicate the
possibilities.
The following table does not define a function from S to G, because not every
member of S is
mapped to a grade:
d  f(d) 
Maria Clara Tom Dick Harry 
A B C B 
The following table does not define a function from S to
G, because one member of S is
mapped to two grades :
d  f(d) 
Maria Clara Tom Dick Harry Harry 
A B C A B NP 
Another way to define a function to specify a rule for how
the function operates , rather than listing
out the mapping. For example, using the common notations
N: the set of natural numbers {1, 2, 3, ...}
Z: the set of all integers {..., 2, 1, 0, 1, 2, ...}
we could define a function f : N → N with the rule f (a) = 2a. The specification
of the domain and
codomain are considered to be part of definition, so the function g : Z → Z
where g(a) = 2a is not
the same function as f, even though the rules are the same .
Note that as in the example above, the range of these functions is not the same
as the codomain.
For example, the range of f is positive even integers, which is not the same as
N. We write
f : N → N to convey the information that everything in the range of f is a
natural number , without
commenting (until we give the rule) on whether every natural number is actually
in the range.
Types of Functions
OnetoOne Functions
A function is called onetoone, or
injective, if it maps distinct elements of the domain to distinct elements of the range. More formally, a function f is onetoone if and only if x ≠ y implies that for f(x) ≠ f(y) for all x and y in the domain of f. Another way to say this is that f(x) = f(y) implies x = y. A function is said to be an injection if it is onetoone. 
For example, the function f: N → N defined by the rule f (x)
= x^{2 }is an injection. However,
the function g: Z → Z defined by the same rule is not an injection, since g(x)
and g(x) both
map to x^{2}.
Onto Functions
A function f: A →B is called onto, or
surjective,
if and only if for every element b in B there is at least one element a in A that maps to it, i.e., there is at least one element a such that f(a) = b. Another way to say this is that the range of f is equal to its target. A function is said to be a surjection if it is onto 
For example, the function f(x) = x + 1 from the set of
integers to the set of integers is onto, since
for every integer y there is an integer x such that f(x) = y. By definition, f(x)
= y if and only if
x + 1 = y, implying that x = y – 1. So, for any y, we can find a value x so that
f(x) = y holds.
The function f(x) = x^{2 }from the set of integers to the set of integers is not
onto. This follows from
the fact that there is no integer x such that f(x) = x^{2 }= –1.
Onetoone Correspondence Functions
A function f is a onetoone correspondence, or
bijective, if it is both onetoone and onto. Such a function is called a bijection. 
The function f(x) = x + 1 from the set of integers to the
set of integers is a bijection. It is oneto
one since x + 1 = y + 1 if and only if x = y. And as we saw above, the function
is onto.
As examples, consider the four functions shown below:
Function a is onetoone and onto (it is a bijection).
Function b is onetoone, but not onto.
Function c is not onetoone, but is onto.
Function d is neither onetoone nor onto.
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