Discrete Mathematics
CS 2610
September 12, 2006
Agenda
Last class

Functions
 Vertical line rule
 Ordered pairs
 Graphical representation
 Predicates as functions
This class

More on functions!
2
Function Terminology
Given a function f:AB

A is the domain of f.

B is the codomain of f.

If f(a)=b, b is the image of a under f.

a is a pre-image of b under f.
 In general, b may have more than 1 pre-image.

The range R of f (or image of f) is :
R={b | a f(a)=b } -- the set of all images

For any set S  A, the image of S,
 f(S) = { b  B | a  S, b = f(a)}

For any set T  B, the inverse image of T
 f−1(T) = { a  A | f(a)  T }
3
Example
Mike
Mario
Kim
Joe
Jill
A
Domain
f
John Smith
Edward Jones
Richard Boone
B
Codomain
The image of Mike under f is John Smith
Mike is a pre-image of John Smith under f
R (f) = {John Smith, Richard Boone}
f(Mike,Mario,Jill) = {John Smith, Richard Boone}
f-1(Richard Boone) = {Joe, Jill}
4
Example
Given a function f: Z  Z where f(x) = x2
-- the domain of f is the set of all integers
-- the codomain of f is the set of all integers
-- the range of f is the set of all integers that are
perfect squares {0, 1, 4, 9, 16, 25, …}
5
Function Composition
Given the functions g:AB and f:BC, the
composition of f and g, f ○g: AC defined as
f ○g (a) = f ( g (a) )
g
h
b
d
o
A
2
f ○g (h) ?
f

3

5

1

7
B
C
6
Function Composition
Properties
Associative: Given the functions g:AB and f:BC
and h:CD then
h ○ (f ○g)  (h ○ f ) ○ g
7
Function Self-Composition
A function f: AA (the domain and codomain are
the same) can be composed with itself
f: People  People
where f(x) is the father of x
f ○f (Mike) is the father of the father of Mike
f ○f ○ f (Mike) ?
f ○f ○ f ○ f(Mike) ?
8
Injective Functions (one-to-one)
A function f: A  B is one-to-one (injective, an
injection) iff f(x) = f(y)  x = y for all x and y in
the domain of f (xy(f(x) = f(y)  x = y))
Equivalently: xy(x  y  f(x)  f(y))
A
f
B
Every b  B has at most 1 pre-image
9
Surjective Functions (onto)
A function f: A  B is onto (surjective, an
surjection)
iff yx( f(x) = y) where y  B, x  A
A
f
B
Every b  B has at least one pre-image
10
Bijective Functions
A function f: A  B is bijective iff it is one-to-one
and onto (a one-to-one correspondence)
f
A
B
The domain cardinality equals the codomain cardinality
11
Inverse Functions
Let f : A  B be a bijection, the inverse of f,
f -1:B  A
such that for any b  B,
f -1(b) = a when f (a) = b
A
f
f-1
B
12
Inverse Functions
Let f: A  B be a bijection, and f-1:B  A be the
inverse of f:
f-1 ○ f = IA = (f-1○f)(a) = f-1 (f(a)) = f-1 (b) = a
f ○ f-1 = IB = (f○f-1)(b) = f(f-1 (b)) = f(a) = b
A
f
f-1
B
13
Functions: Real Functions
Given f :RR and g :RR then

(f  g): RR, is defined as
(f  g)(x) = f(x)  g(x)

(f . g): RR is defined as
(f g)(x) = f(x) × g(x)
Example:
Let f :RR be f(x) = 2x and g :RR be g(x) = x3
(f+g)(x) = x3+2x
(f . g)(x) = 2x4
14