CSCI 115
Chapter 5
Functions
CSCI 115
§5.1
Functions
§5.1 – Functions
• Function
– Relation such that for every domain element a,
|f(a)| = 1
– Mappings, transformations
– f(a) = {b}
• f(a) = b
• a = argument
• b = function value
§5.1 – Functions
• Types of functions
–
–
–
–
Everywhere defined
Onto
One to one (1–1)
1 – 1 Correspondence
• ED, Onto, 1–1
– Invertible Functions
§5.1 – Functions
• Theorem 5.1.1
Let f : A  B be any function.
(a) Then f 1 is a function from B to A iff f is 1 - 1.
If f 1 is a function, then
(b) f 1 is also 1 - 1
(c) f 1 is everywhere defined iff f is onto
(d) f 1 is onto iff f is everywhere defined
§5.1 – Functions
• Theorem 5.1.2
Let f : A  B be any function. Then
(a) 1B  f  f
(b) f  1A  f
If f is a 1 - 1 correspond ence between A and B then
(c) f 1  f  1A
(d) f  f 1  1B
§5.1 – Functions
• Theorem 5.1.4
Let A and B be two finite sets with the same number
of elements, and let f : A  B be an everywhere
defined function.
(a) If f is 1 - 1, then f is onto.
(b) If f is onto, then f is 1 - 1.
§5.1 – Functions
• 1-1 functions and cryptography
– Allows coding AND decoding
– Substitution codes
• Table from Example 18 (p. 188)
CSCI 115
§5.2
Functions for Computer Science
§5.2 – Functions for Computer
Science
•
•
•
•
•
•
•
mod–n (or modn)
Factorial
Floor
Ceiling
Boolean
Hashing
Others
§5.2 – Functions for Computer
Science
• Set
– Collection of objects
– Any element is unambiguously in the set or not
– Characteristic function
• Fuzzy sets
– Whether or not an element is in the set may be
‘fuzzy’
– The set of all rich people
§5.2 – Functions for Computer
Science
• Fuzzy sets
– Function f defined on a set having values in the
interval [0, 1]
• If f(x) = 0, x is not in the set
• If f(x) = 1, x is in the set
• If 0 < f(x) < 1 then f(x) is the degree to which x is
in the set
– Ordinary sets are special cases of fuzzy sets
§5.2 – Functions for Computer
Science
• Fuzzy set operations
– Theorem 5.2.1
• Let A and B be subsets of the same universal set U. Then:
(a ) f A B ( x)  min{ f a ( x), f b ( x )}
(b) f A B ( x)  max{ f a ( x ), f b ( x )}
(c ) f A ( x )  1  f A ( x )
• Do problems #30, 34, 36 on p. 198 – 199
§5.2 – Functions for Computer
Science
• Fuzzy Logic
– Fuzzy predicates
• Values can be true, false, or somewhere in between
– Schrodinger’s Cat
– Quantum theory and computers
– Applications
• Control theory
– Elevator operation
– ABS systems in cars
• Expert systems
CSCI 115
§5.3
Growth of Functions
§5.3 – Growth of Functions
• Algorithmic Analysis
– Efficiency
• Number of steps (running time)
– Comparison
§5.3 – Growth of Functions
• Definitions
– Let f and g be functions whose domains are
subsets of Z+.
• We say f is O(g) (read f is big-Oh of g) if 
constants c and k s.t. |f(n)|  c |g(n)|  n  k.
• We say f and g have the same order if f is O(g) and
g is O(f)
• We say f is lower order than g (or f grows more
slowly than g) if f is O(g) but g is not O(f)
§5.3 – Growth of Functions
• Definition
– We define a relation Θ (called big-theta) on
functions whose domains are subsets of Z+ as:
f Θ g iff f and g have the same order.
• Theorem 5.3.1
– The relation Θ is an equivalence relation.
§5.3 – Growth of Functions
• Equivalence classes of Θ
– Equivalence classes (called Θ–classes) consist
of functions of the same order
– One Θ–class is said to be lower than another if
any of the functions in the first is lower than
any in the second
– Θ–classes provide the necessary information to
do algorithmic analysis
§5.3 – Growth of Functions
•
Image from page 203 of the text
CSCI 115
§5.4
Permutation Functions
§5.4 – Permutation Functions
• Permutation
– 1-1 correspondence from a set onto itself
• Theorem 5.4.1
– If A = {a1, a2, …, an} with |A| = n, then n!
permutations of A
§5.4 – Permutation Functions
• Cycle of length r
– If p(a1) = a2, p(a2) = a3, …, p(ar-1) = ar and
p(ar) = a1, this is called a cycle of length r, and
is denoted (a1, a2,…, ar)
• Disjoint cycles
§5.4 – Permutation Functions
• Theorem 5.4.2
– A permutation of a finite set that is not the identity
or a cycle can be written as a product of disjoint
cycles of length greater than or equal to 2
§5.4 – Permutation Functions
• Transposition
– Cycle of length 2
– Every cycle can be written as a product of
transpositions as follows:
• (b1, b2, …, br) = (b1, br)(b1, br-1)  …  (b1, b2)
§5.4 – Permutation Functions
• Even permutation
– A permutation that can be written as the
product of an even number of transpositions
• Odd permutation
– A permutation that can be written as the
product of an odd number of transpositions
§5.4 – Permutation Functions
• Theorem 5.4.3
– A permutation cannot be both even and odd
• Theorem 5.4.4
– Let A = {a1, a2, …, an}, with |A| = n  2. Then
there are n!/2 even permutations of A, and n!/2
odd permutations of A.
§5.4 – Permutation Functions
• Transposition codes
– Encrypt by using the following transposition
MATH IS GREAT
(1, 10, 7)  (11, 3, 2, 8)  (5, 4, 9)
– Decrypt the following
Message: OICLPCYIGULSOYRRVTTEA
Transposition
Code:
(16, 15, 6, 1, 3, 7, 11, 2)  (8, 13, 4, 5, 19)  (14, 10)  (21, 20, 17)
§5.4 – Permutation Functions
• Transposition codes
– Keyword Columnar Transposition – Example 8
Using the keyword “JONES” to encrypt:
THE FIFTH GOBLET CONTAINS THE GOLD
Result:
FGTAHDTFBONGEHETTLHTLNSOIOCIEX
§5.4 – Permutation Functions
• Transposition codes
– Keyword Columnar Transposition
Use the keyword “BASEBALL” to decrypt:
AAK7ENWHRSOER9SAETOELNNWIDYELXBO1DXKTI3R
Using a keyword columnar transposition