1. Discrete Math for Computer Science CSC 281
Dr.Yuan Tian
Module Topic
Basic Structures: Functions

2. Acknowledgement
Most of these slides were either created by Professor Bart Selman at Cornell University or else are modifications of his slides

3. Functions f(x) x f(x) = -(1/2)x – 1/2 Suppose we have:
How do you describe the yellow function?
f(x) = -(1/2)x – 1/2
What’s a function ?

4. Functions Kathy Michael Toby John Chris Brad Carol Mary B A
A = {Michael, Toby , John , Chris , Brad }
B = { Kathy, Carla, Mary}
Let f: A  B be defined as f(a) = mother(a).
Michael Toby John Chris Brad
Kathy
Carol
Mary A B

5. Functions
A function f from a set A to a set B is an assignment of exactly one element of B to each element of A.
We write
f(a) = b
if b is the unique element of B assigned by the function f to the element a of A.
If f is a function from A to B, we write
f: AB
(note: Here, ““ has nothing to do with if… then)
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6. Functions Note: Functions are also called mappings or transformations.
More generally: B
Definition:
Given A and B, nonempty sets, a function f from A to B is an assignment
of exactly one element of B to each element of A. We write f(a)=b if b is
the element of B assigned by function f to the element a of A.
If f is a function from A to B, we write f : AB.
Note: Functions are also called mappings or transformations.

7. Functions A - Domain of f B- Co-Domain of f domain More generally:
f: RR, f(x) = -(1/2)x – 1/2
domain
co-domain

8. Functions
If f:AB, we say that A is the domain of f and B is the codomain of f.
If f(a) = b, we say that b is the image of a and a is the pre-image of b.
The range of f:AB is the set of all images of elements of A.
We say that f:AB maps A to B.
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9. Functions a collection of points! a point! Why not?
More formally: a function f : A  B is a subset of AxB where  a  A, ! b  B and <a,b>  f.
a point! A B B A
Why not?

10. Functions Let us take a look at the function f:PC with
P = {Linda, Max, Kathy, Peter}
C = {Boston, New York, Hong Kong, Moscow}
f(Linda) = Moscow
f(Max) = Boston
f(Kathy) = Hong Kong
f(Peter) = New York
Here, the range of f is C.
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11. Functions Let us re-specify f as follows: f(Linda) = Moscow
f(Max) = Boston
f(Kathy) = Hong Kong
f(Peter) = Boston
Is f still a function?
yes
What is its range?
{Moscow, Boston, Hong Kong}
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12. Functions
If the domain of our function f is large, it is convenient to specify f with a formula, e.g.:
f:RR
f(x) = 2x
This leads to:
f(1) = 2
f(3) = 6
f(-3) = -6 …
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13. Functions Other ways to represent f: Boston Peter Hong Kong Kathy Max
Moscow
Linda
f(x) x
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow
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14. Functions Let f1 and f2 be functions from A to R.
Then the sum and the product of f1 and f2 are also functions from A to R defined by:
(f1 + f2)(x) = f1(x) + f2(x)
(f1f2)(x) = f1(x) f2(x)
Example:
f1(x) = 3x, f2(x) = x + 5
(f1 + f2)(x) = f1(x) + f2(x) = 3x + x + 5 = 4x + 5
(f1f2)(x) = f1(x) f2(x) = 3x (x + 5) = 3x2 + 15x
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15. Functions
We already know that the range of a function f:AB is the set of all images of elements aA.
If we only regard a subset SA, the set of all images of elements sS is called the image of S.
We denote the image of S by f(S):
f(S) = {f(s) | sS}
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16. Functions Let us look at the following well-known function:
f(Linda) = Moscow
f(Max) = Boston
f(Kathy) = Hong Kong
f(Peter) = Boston
What is the image of S = {Linda, Max} ?
f(S) = {Moscow, Boston}
What is the image of S = {Max, Peter} ?
f(S) = {Boston}
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17. Functions - image & preimage
image(S)
For any set S  A, image(S) = {b : a  S, f(a) = b}
So, image({Michael, Toby}) = {Kathy} image(A) = B - {Carol}
range of f
image(A)
Michael Toby John Chris Brad
Kathy
Carol
Mary B A
image(John) = {Kathy}
pre-image(Kathy) = {John, Toby, Michael}

18. Properties of Functions
A function f:AB is said to be one-to-one (or injective), if and only if
x, yA (f(x) = f(y)  x = y)
In other words: f is one-to-one if and only if it does not map two distinct elements of A onto the same element of B.
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19. Properties of Functions
g(Linda) = Moscow
g(Max) = Boston
g(Kathy) = Hong Kong
g(Peter) = New York
Is g one-to-one?
Yes, each element is assigned a unique element of the image.
And again…
f(Linda) = Moscow
f(Max) = Boston
f(Kathy) = Hong Kong
f(Peter) = Boston
Is f one-to-one?
No, Max and Peter are mapped onto the same element of the image.
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20. Every b  B has at most 1 preimage.
Functions - injection
A function f: A  B is one-to-one (injective, an injection) if a,b,c, (f(a) = b  f(c) = b)  a = c
Not one-to-one
Michael Toby John Chris Brad
Kathy
Carol
Mary

21. Properties of Functions
How can we prove that a function f is one-to-one?
Whenever you want to prove something, first take a look at the relevant definition(s):
x, yA (f(x) = f(y)  x = y)
Example:
f:RR
f(x) = x2
Disproof by counterexample:
f(3) = f(-3), but 3  -3, so f is not one-to-one.
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22. Properties of Functions
… and yet another example:
f:RR
f(x) = 3x
One-to-one: x, yA (f(x) = f(y)  x = y)
To show: f(x)  f(y) whenever x  y
x  y
3x  3y
f(x)  f(y),
so if x  y, then f(x)  f(y), that is, f is one-to-one.
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23. Properties of Functions
A function f:AB with A,B  R is called strictly increasing, if
x,yA (x < y  f(x) < f(y)),
and strictly decreasing, if
x,yA (x < y  f(x) > f(y)).
Obviously, a function that is either strictly increasing or strictly decreasing is one-to-one.
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24. Properties of Functions
A function f:AB is called onto, or surjective, if and only if for every element bB there is an element aA with f(a) = b.
In other words, f is onto if and only if its range is its entire codomain.
A function f: AB is a one-to-one correspondence, or a bijection, if and only if it is both one-to-one and onto.
Obviously, if f is a bijection and A and B are finite sets, then |A| = |B|.
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25. Functions - surjection
Every b  B has at least 1 preimage.
Functions - surjection
A function f: A  B is onto (surjective, a surjection) if b  B, a  A f(a) = b
Not onto
Michael Toby John Chris Brad
Kathy
Carol
Mary

26. Functions – one-to-one-correspondence or bijection
A function f: A  B is bijective if it is one-to-one and onto.
Every b  B has exactly 1 preimage.
Anna Mark John
Paul
Sarah
Carol Jo Martha Dawn
Eve
Anna Mark John Paul Sarah
Carol Jo
Martha
Dawn
Eve
An important implication of this characteristic:
The preimage (f-1) is a function!
They are
invertible.

27. Properties of Functions
Examples:
In the following examples, we use the arrow representation to illustrate functions f:AB.
In each example, the complete sets A and B are shown.
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28. Properties of Functions
Is f injective?
No.
Is f surjective?
Is f bijective?
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow
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29. Properties of Functions
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow
Is f injective?
No.
Is f surjective?
Yes.
Is f bijective?
Paul
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30. Properties of Functions
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow
Lübeck
Is f injective?
Yes.
Is f surjective?
No.
Is f bijective?
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31. Properties of Functions
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow
Lübeck
Is f injective?
No! f is not even a function!
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32. Properties of Functions
Linda
Boston
Is f injective?
Yes.
Is f surjective?
Is f bijective?
Max
New York
Kathy
Hong Kong
Peter
Moscow
Helena
Lübeck
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33. Functions: inverse function
Definition:
Given f, a one-to-one correspondence from set A to set B, the inverse
function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a)=b. The inverse function is denoted f-1 . f-1 (b)=a, when f(a)=b. B

34. Inversion
An interesting property of bijections is that they have an inverse function.
The inverse function of the bijection f:AB is the function f-1:BA with
f-1(b) = a whenever f(a) = b.
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35. Inversion Example: f(Linda) = Moscow f(Max) = Boston
f(Kathy) = Hong Kong
f(Peter) = Lübeck
f(Helena) = New York
Clearly, f is bijective.
The inverse function f-1 is given by:
f-1(Moscow) = Linda
f-1(Boston) = Max
f-1(Hong Kong) = Kathy
f-1(Lübeck) = Peter
f-1(New York) = Helena
Inversion is only possible for bijections (= invertible functions)
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36. Inversion Linda Boston f Max New York f-1
f-1:CP is no function, because it is not defined for all elements of C and assigns two images to the pre-image New York.
Kathy
Hong Kong
Peter
Moscow
Helena
Lübeck
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37. Functions - examples yes yes yes This function is invertible.
Suppose f: R+  R+, f(x) = x2.
Is f one-to-one?
Is f onto?
Is f bijective?
yes
yes
yes
This function is invertible.

38. Functions - examples no yes no This function is not invertible.
Suppose f: R  R+, f(x) = x2.
Is f one-to-one?
Is f onto?
Is f bijective? no
yes no
This function is not invertible.

39. Functions - examples no no no Suppose f: R  R, f(x) = x2.
Is f one-to-one?
Is f onto?
Is f bijective? no no no

41. Functions - composition
“f composed with g”
Let f: AB, and g: BC be functions. Then the composition of f and g is:
(f o g)(x) = f(g(x))
Note: (f o g) cannot be defined unless the range of g is a subset of the domain of f.

42. Composition
The composition of two functions g:AB and f:BC, denoted by fg, is defined by
(fg)(a) = f(g(a))
This means that
first, function g is applied to element aA, mapping it onto an element of B,
then, function f is applied to this element of B, mapping it onto an element of C.
Therefore, the composite function maps from A to C.
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43. Composition Example: f(x) = 7x – 4, g(x) = 3x, f:RR, g:RR
(fg)(5) = f(g(5)) = f(15) = 105 – 4 = 101
(fg)(x) = f(g(x)) = f(3x) = 21x - 4
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44. Composition Composition of a function and its inverse:
(f-1f)(x) = f-1(f(x)) = x
The composition of a function and its inverse is the identity function i(x) = x.
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45. Composition Example: Let f(x) = 2 x +3; g(x) = 3 x + 2;
(f o g) (x) = f(3x + 2) = 2 (3 x + 2 ) + 3 = 6 x + 7.
(g o f ) (x) = g (2 x + 3) = 3 (2 x + 3) + 2 = 6 x + 11.
As this example shows, (f o g) and (g o f) are not necessarily equal – i.e, the composition of functions is not commutative.

46. Composition Note: (f -1 o f) (a) = f -1(f(a)) = f -1(b) = a.
(f o f -1) (b) = f (f -1(b)) = f-(a) = b.
Therefore (f-1o f ) = IA and (f o f-1) = IB where IA and IB are the identity
function on the sets A and B. (f -1) -1= f

47. Some important functions
Absolute value:
Domain R; Co-Domain = {0}  R+
|x| =
x if x ≥0
-x if x < 0
Ex: |-3| = 3; |3| = 3
Floor function (or greatest integer function):
Domain = R; Co-Domain = Z
x  = largest integer not greater than x
Ex: 3.2 = 3; -2.5 =-3

48. Some important functions
Ceiling function:
Domain = R;
Co-Domain = Z
x = smallest integer greater than x
Ex: 3.2 = 4; -2.5 =-2

49. ≤ +

50. Floor and Ceiling Functions
The floor and ceiling functions map the real numbers onto the integers (RZ).
The floor function assigns to rR the largest zZ with z  r, denoted by r.
Examples: 2.3 = 2, 2 = 2, 0.5 = 0, -3.5 = -4
The ceiling function assigns to rR the smallest zZ with z  r, denoted by r.
Examples: 2.3 = 3, 2 = 2, 0.5 = 1, -3.5 = -3
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51. Some important functions
Factorial function: Domain = Range = N Error on range
n! = n (n-1)(n-2) …, 3 x 2 x 1
Ex: 5! = 5 x 4 x 3 x 2 x 1 = 120
Note: 0! = 1 by convention.

52. Some important functions
Mod (or remainder):
Domain = N x N+ = {(m,n)| m N, n  N+ }
Co-domain Range = N
m mod n = m - m/n n
Ex: 8 mod 3 = 8 - 8/3 3 = 2
57 mod 12 = 9;
Note: This function computes the remainder when m is divided by n.
The name of this function is an abbreviation of m modulo n, where modulus means with
respect to a modulus (size) of n, which is defined to be the remainder when m is divided
by n. Note also that this function is an example in which the domain of the function is a
2-tuple.

53. Some important functions: Exponential Function
Domain = R+ x R = {(a,x)| a  R+, x  R }
Co-domain Range = R+
f(x) = a x
Note: a is a positive constant; x varies.
Ex: f(n) = a n = a x a …, x a (n times)
How do we define f(x) if x is not a positive integer?

54. Some important functions: Exponential function
How do we define f(x) if x is not a positive integer?
Important properties of exponential functions:
a (x+y) = ax ay; (2) a 1 = a (3) a 0 = 1
See:

55. We get: By similar arguments:
Note: This determines ax for all x rational. x is irrational by continuity (we’ll skip “details”).

56. Some important functions: Logarithm Function
Logarithm base a:
Domain = R+ x R = {(a,x)| a  R+, a>1, x  R }
Co-domain Range = R
y = log a (x)  ay = x
Ex: log 2 (8) =3; log 2 (16) =3; 3 < log 2 (15) <4.
Key properties of the log function (they follow from those for exponential):
log a (1)=0 (because a0 =1)
log a (a)=1 (because a1 =a)
log a (xy) = log a (x) + log a (x) (similar arguments)
log a (xr) = r log a (x)
log a (1/x) = - log a (x) (note 1/x = x-1)
log b (x) = log a (x) / log a (b)

57. Logarithm Functions log 2 (1/4)= - log 2 (4)= - 2.
Examples:
log 2 (1/4)= - log 2 (4)= - 2.
log 2 (-4) undefined
log 2 ( )= log 2 (210) + log 2 (35 )=10 log 2 (2) + 5log 2 (3 )=
= log 2 (3 )

58. Limit Properties of Log Function
As x gets large, log(x) grows without bound.
But x grows MUCH faster than log(x)…more soon on growth rates.

59. Some important functions: Polynomials
Polynomial function:
Domain = usually R
Co-domain Range = usually R
Pn(x) = anxn + an-1xn-1 + … + a1x1 + a0
n, a nonnegative integer is the degree of the polynomial;
an 0 (so that the term anxn actually appears)
(an, an-1, …, a1, a0) are the coefficients of the polynomial.
Ex:
y = P1(x) = a1x1 + a0 linear function
y = P2(x) = a2x2 + a1x1 + a0 quadratic polynomial or function

60. We’ll talk more about growth rates in the next module….
Exponentials grow MUCH faster than polynomials:
We’ll talk more about growth rates in the next module….