Generic Functions - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

Download Generic Functions - Discrete Mathematics - Lecture Slides and more Slides Discrete Mathematics in PDF only on Docsity!

Discrete Mathematics

Lecture 7

Generic Functions

• A function f: X  Y is a relationship between elements of X to elements of Y, when each element from X is related to a unique element from Y
• X is called domain of f, range of f is a subset of Y so that for each element y of this subset there exists an element x from X such that y = f(x)
• Sample functions:
  • f : R  R, f(x) = x^2
  • f : Z  Z, f(x) = x + 1
  • f : Q  Z, f(x) = 2

Generic Functions

• Encoding and decoding of characters
• Hamming distance function: number of

differences between two encodings

• Boolean functions: f : {0, 1}n^  {0, 1}
• Well-defined functions
• Euler function: ϕ(n) is the number of

positive integers less than n, which are

mutually prime with n

Exercises

• Show that if p is a prime number, then ϕ(p n) = p n^ – p n-
• Prove that there infinitely many integers for which Euler’s function is a perfect square
• Show that ϕ(pq) = ϕ(p-1) ϕ(q-1) if p and q are distinct primes
• Determine which of the following is true:
  • if A ⊆ B, then f(A) ⊆ f(B)
  • f(A ∪ B) = f(A) ∪ f(B)
  • f(A ∩ B) = f(A) ∩ f(B)
  • f(A - B) = f(A) - f(B)

FSA by Transition Diagram

s0 s1 s

1

1

1

0

0 0

FSA by Next-State Table

a b c

U Z Y Y

X V V V

Y Z V Y

Z Z Z Z

Designing FSA

• Design an FSA that accepts all strings of 0’s and

1’s such that the number of 1’s is divisible by 3

• Design an FSA that accepts the set of strings that

contain exactly one 1

• Design an FSA with alphabet {a, b} which accepts

strings that end on the same two characters

• Simulating an FSA using software

One-to-One Functions

• Function f : X  Y is called one-to-one

(injective) when for all elements x 1 and x 2

from X if f(x 1 ) = f(x 2 ), then x 1 = x 2

• Determine whether the following functions

are one-to-one:

• f : R  R, f(x) = 4x – 1
• g : Z  Z, g(n) = n 2
• Hash functions

Inverse Functions

• If f : X  Y is a bijective function, then it is

possible to define an inverse function f-1^ : Y

 X so that f -1^ (y) = x whenever f(x) = y

• Find an inverse for the following functions:
  • String-reverse function
  • f : R  R, f(x) = 4x – 1
• Inverse function of a bijective function is a

bijective function itself

Exercises

• Let c (^) m,n be the number of onto functions

from a set of m elements to a set of n

elements. Find a relationship between cm,n,

c m-1,n and c m-1,n-

• Let F: Z × Z  Z and G: Z × Z  Z, F(n,

m) = 3 n 6 m^ and G(n, m) = 3 n 5 m. Is F one-to-

one, is G one-to-one?

Pigeonhole Principle

• There is no FSA that accepts the following language: L = {s = akb k, for positive k}
• Generalized Pigeonhole Principle: For any function f : X  Y acting on finite sets, if n(X) > k * N(Y), then there exists some y from Y so that there are at least k + 1 distinct x’s so that f(x) = y
• There are 42 students who are to share 12 computers. Each student uses exactly 1 computer and no computer is used by more than 6 students. Show that at least 5 computers are used by 3 or more students.

Exercises

• Let f : X  Y and n(X) = n(Y), then f is bijective iff f is surjective
• Let A be a set of 6 integers less than 13. Show that there must be two disjoint subsets of A whose sum of elements adds up to the same number
• Given 52 distinct integers, show that there must be two whose sum or difference is divisible by 100
• Show that if 101 integers are chosen from 1 to 200 inclusive, there must be two with the property that one is divisible by the other
• Suppose a 1 , a 2 , …, a (^) n is a sequence of n integers none of which is divisible by n. Show that at least one difference a (^) i
  • a (^) j is divisible by n

Cardinality

• Cardinality refers to the size of the set
• Finite and infinite sets
• Two sets have the same cardinality when there is bijective function associating them
• Cardinality is is reflexive, symmetric and transitive
• Countable sets: set of all integers, set of even numbers
• Set of real numbers
• Computability of functions

Exercises

• Show that the set of irrational numbers is

dense

• Show that a power set has always a greater

cardinality than the original set