Cpsc125 Ch4 Sec4

Relations, Functions, and Matrices

Mathematical
Structures for
Computer Science
Chapter 4

Section 4.4 © 2006 W.H. Freeman & Co.
Copyright MSCS Slides

Functions Functions
Tuesday, March 23, 2010

Functions
● DEFINITIONS: TERMINOLOGY FOR FUNCTIONS
Let S and T be sets. A function (mapping) f from S to T, f :
S → T, is a subset of S × T, where each member of S
appears exactly once as the ﬁrst component of an ordered
pair. S is the domain and T the codomain of the function.
If (s,t) belongs to the function, then t is denoted by f(s); t is
the image of s under f, s is a preimage of t under f, and f is
said to map s to t. For A ⊆ S, f (A) denotes { f (a) ⏐ a ∈ A}.
● There are three parts to a function:
A set of starting values
■

A set from which associated values come
■

The association itself
■

Section 4.4 Functions 2

Functions
● The set of starting values is called the domain of the
function.
● The set from which associated values come is called
the codomain of the function.
● Here f is a function from S to T, symbolized f: S → T. S is
the domain and T is the codomain. The association itself is
a set of ordered pairs, each of the form (s,t) where s ∈ S, t
∈ T, and t is the value from T that the function associates
with the value s from S; t = f (s).


Functions
● A function from S to T is a subset of S × T with certain
restrictions on the ordered pairs it contains.
■
By the definition of a function, a binary relation that is one-to-
many (or many-to-many) cannot be a function.
■
Each member of S must be used as a first component.
● The definition of a function includes functions of more than one
variable. We can have a function f : S1 × S2 × ... × Sn → T that
associates with each ordered n-tuple of elements (s1, s2, ... , sn), si
∈ Si, a unique element of T.
● The floor function ⎣x⎦ associates with each real number x the
greatest integer less than or equal to x.
● The ceiling function ⎡x⎤ associates with each real number x the
smallest integer greater than or equal to x.
● ⎣2.8⎦ = 2, ⎡2.8⎤ = 3, 4. Both the floor function and the ceiling
function are functions from R to Z.


Functions

● For any integer x and any positive integer n, the modulo
function, denoted by f (x) = x mod n, associates with x the
remainder when x is divided by n. One can write x as x =
qn + r, 0 ≤ r ≤ n, where q is the quotient and r is the
remainder, so the value of x mod n is r.
● Not all functional associations can be described by an
equation. Technically, the equation only describes a way to
compute associated values.
● g: R → R, where g(x) = x3.
● f : Z → R, given by f (x) = x3 is not the same function as g.
The domain has been changed, which changes the set of
ordered pairs.


Functions
● DEFINITION: EQUAL FUNCTIONS
Two functions are equal if they have the same
domain, the same codomain, and the same association
of values of the codomain with values of the domain.
● To show that two functions with the same domain and
the same codomain are equal, one must show that the
associations are the same.
● This can be done by showing that, given an arbitrary
element of the domain, both functions produce the
same associated value for that element; that is, they
map it to the same place.


Onto Functions
● DEFINITION: ONTO (SURJECTIVE) FUNCTION
A function f: S → T is an onto, or surjective, function if
the range of f equals the codomain of f.
● In every function with range R and codomain T, R ⊆ T.
● To prove that a given function is onto,
● Show that T ⊆ R; then it will be true that R = T.
● Show that an arbitrary member of the codomain is a
member of the range.
● g: R → R where g(x) = x3 is an onto function.


One-to-One Functions
● DEFINITION: ONE-TO-ONE (INJECTIVE)
FUNCTION A function f: S → T is one-to-one, or
injective, if no member of T is the image under f of two
distinct elements of S.
● The one-to-one idea here is the same as for binary
relations in general, except that every element of S
must appear as a ﬁrst component in an ordered pair.
● To prove that a function is one-to-one, we assume that
there are elements s1 and s2 of S with f (s1) = f (s2) and
then show that s1 = s2.
● The function g: R → R deﬁned by g(x) = x3 is one-to-
one because if x and y are real numbers with g(x) = g
(y), then x3 = y3 and x = y.

Bijections
● DEFINITION: BIJECTIVE FUNCTION
A function f:S → T is bijective (a bijection) if it is both
one-to-one and onto.
● The function g: R → R given by g(x) = x3 is a bijection.


Composition of Functions
● DEFINITION: COMPOSITION FUNCTION
Let f: S → T and g: T → U. Then the composition function, g ° f,
is a function from S to U deﬁned by (g ° f )(s) = g( f (s)).
● The function g ° f is applied right to left; function f is applied ﬁrst
and then function g.
● Function composition preserves the properties of being onto and
being one-to-one.
● THEOREM ON COMPOSING TWO BIJECTIONS The
composition of two bijections is a bijection.


Inverse Functions
● Let f: S T be a bijection. Because f is onto, every t ∈ T
has a preimage in S. Because f is one-to-one, that preimage
is unique.
● The function that maps each element of a set S to itself,
that is, that leaves each element of S unchanged, is called
the identity function on S and denoted by iS.
● DEFINITION: INVERSE FUNCTION Let f be a
function, f: S → T. If there exists a function g: T → S such
that g ° f = iS and f ° g = iT, then g is called the inverse
function of f, denoted by f -1.
● THEOREM ON BIJECTIONS AND INVERSE
FUNCTIONS
Let f: S T. Then f is a bijection if and only if f -1 exists.


Permutation Functions
● DEFINITION: PERMUTATIONS OF A SET For a
given set A, SA = { f ⏐ f: A → A and f is a bijection}.
SA is thus the set of all bijections of set A into (and
therefore onto) itself; such functions are called
permutations of A.
● If f and g both belong to SA, then they each have
domain = range = A.
● If A = {1, 2, 3, 4}, one permutation function of A, call
it f, is given by f = {(1,2), (2,3), (3,1), (4,4)}.
● A shorter way to describe the permutation f is to use
cycle notation and write f = (1, 2, 3), understood to
mean that f maps each element listed to the one on its
right, the last element listed to the ﬁrst, and an element
of the domain not listed to itself.

Permutation Functions

● If we were to compute f ° g (1, 2, 3) ° (2,3), we would
get (1,2).
● If, however, f and g are members of SA and f and g are
disjoint cycles—the cycles have no elements in
common—then f ° g = g ° f.
● The permutation that maps each element of A to itself
is the identity function on A, iA, also called the
identity permutation.
● A permutation on a set that maps no element to itself
is called a derangement.


How Many Functions?
● THEOREM ON THE NUMBER OF FUNCTIONS
WITH FINITE DOMAINS AND CODOMAINS

If ⏐S⏐ = m and ⏐T⏐ = n, then:
1. The number of functions f: S → T is nm.
! The number of one-to-one functions f: S → T, assuming m
> n, is n!/(n − m)!
! The number of onto functions f: S → T, assuming m ≤ n, is
nm − C(n, 1)(n − 1)m + C(n, 2)(n − 2)m − C(n, 3)(n − 3)m
+...+ (−1)n − 1C(n, n − 1)(1)m

● For example, let S = {A, B, C} and T = {a, b}. Find the
number of functions from S onto T.
● 23 − C(2, 1)(1)3 = 8 − 2 • 1 = 6


How Many Functions?
● If A is a set with ⏐A⏐= n, then the number of permutations
of A is n!
● This number can be obtained by any of three methods:
■
A combinatorial argument (each of the n elements in the
domain must map to one of the n elements in the range with
no repetitions)
■
Thinking of such functions as permutations on a set with n
elements and noting that P(n,n) = n!
■
Using result (2) in the previous theorem with m = n


Equivalent Sets
● DEFINITIONS: EQUIVALENT SETS AND
CARDINALITY
A set S is equivalent to a set T if there exists a bijection f:
S → T. Two sets that are equivalent have the same
cardinality.
● The notion of equivalent sets allows us to extend our
definition of cardinality from finite to infinite sets.
● If S is equivalent to T, then all the members of S and T
are paired off by f in a one-to-one correspondence.
● CANTOR’S THEOREM
For any set S, S and ℘(S) are not equivalent.


Order of Magnitude of Functions

● Order of magnitude is a way of comparing the “rate of
growth” of different functions.
● For instance, if we compute f (x) = x and g(x) = x2 for
increasing values of x, the g-values will be larger than the
f-values by an ever increasing amount.
● This difference in the rate of increase cannot be overcome
by simply multiplying the f-values by some large constant;
● DEFINITION: ORDER OF MAGNITUDE
Let f and g be functions mapping nonnegative reals into
nonnegative reals. Then f is the same order of magnitude
as g, written f=Θ(g), if there exist positive constants n0, c1,
and c2 such that for x ≥ n0, c1 g(x) ≤ f (x) ≤ c2 g(x).



● For example, say f = Θ(x2) and g =Θ(x2). A
polynomial is always the order of magnitude of its
highest-degree term; lower-order terms and all
coefﬁcients can be ignored.
● Order of magnitude is important in analysis of
algorithms.
● Usually the number of times such tasks must be done
in executing the algorithm will depend on the size of
the input.
● Rather than compute the exact functions for the
amount of work done, it is easier and often just as
useful to settle for order-of-magnitude information.



● DEFINITION: BIG OH
Let f and g be functions mapping nonnegative reals
into nonnegative reals. Then f is big oh of g, written f
= O(g), if there exist positive constants n0 and c such
that for x ≥ n0, f (x) ≤ cg(x).
● If f (n) represents the work done by an algorithm on an
input of size n, it may be difficult to find a simple
function g such that f = Θ(g).
● We may still be able to find a function g that serves as
an upper bound for f. In other words, while f may not
have the same shape as g, f will never grow
significantly faster than g.
The big oh notation f = O(g) says that f grows at the
same rate or at a slower rate than g.


● If we know that f deﬁnitely grows at a slower rate than
g, then we can say something stronger.
● This is the little oh of g, written f = o(g). The
relationship between big oh and little oh is this: If f =
O(g), then either f= Θ(g) or f = o(g).


Cpsc125 Ch4 Sec4

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Cpsc125 Ch4 Sec4