discrete mathematics cs 2610 september 12, 2006. 2 agenda last class functions vertical line rule ...
TRANSCRIPT
Discrete Mathematics CS 2610
September 12, 2006
2
Agenda
Last class Functions
Vertical line rule Ordered pairs Graphical representation Predicates as functions
This class More on functions!
3
Function TerminologyGiven a function f:AB
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 }
4
Example
Mike Mario
Kim
Joe Jill
John Smith
Edward Jones
Richard Boone
f
A BDomain Codomain
The image of Mike under f is John SmithMike 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}
5
ExampleGiven 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, …}
6
Function Composition
Given the functions g:AB and f:BC, the composition of f and g, f ○g: AC defined as
f ○g (a) = f ( g (a) )
h
b
d
o
2
3
5
1
7
fg
A B C
f ○g (h) ?
7
Function Composition
Properties
Associative: Given the functions g:AB and f:BC and h:CD then
h ○ (f ○g) (h ○ f ) ○ g
8
Function Self-Composition
A function f: AA (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) ?
9
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 (xy(f(x) = f(y) x = y))
Equivalently: xy(x y f(x) f(y))
Every b B has at most 1 pre-image
fA B
10
Surjective Functions (onto)
A function f: A B is onto (surjective, an surjection)
iff yx( f(x) = y) where y B, x A
Every b B has at least one pre-image
fA B
11
Bijective Functions
A function f: A B is bijective iff it is one-to-one and onto (a one-to-one correspondence)
f
The domain cardinality equals the codomain cardinality
A B
12
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 Bf
f-1
13
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 Bf
f-1
14
Functions: Real Functions
Given f :RR and g :RR then
(f g): RR, is defined as
(f g)(x) = f(x) g(x)
(f . g): RR is defined as
(f g)(x) = f(x) × g(x)
Example:
Let f :RR be f(x) = 2x and g :RR be g(x) = x3
(f+g)(x) = x3+2x
(f . g)(x) = 2x4