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Discrete Mathematics Lecture 7PowerPoint Presentation

Discrete Mathematics Lecture 7

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Discrete Mathematics Lecture 7. Alexander Bukharovich New York University. 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

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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) = x2
- f : Z Z, f(x) = x + 1
- f : Q Z, f(x) = 2

Generic Functions

- Arrow diagrams for functions
- Non-functions
- Equality of functions:
- f(x) = |x| and g(x) = sqrt(x2)

- Identity function
- Sequences as functions
- Functions with a domain of some language
- Logarithmic function

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: (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 (pn) = pn – pn-1
- Prove that there infinitely many integers for which Euler’s function is a perfect square
- Show that (pq) = (p-1) (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)

Finite-State Automata (FSA)

- Finite-state automata A is defined by 5 objects:
- Set I of the input alphabet
- Set S of automaton states
- Designated initial state s0 from S
- Designated set of accepted states from S
- Next-state function N: S I S that associates next state to the pair {current state, input symbol}

- Descriptions of finite-state automaton:
- State-transition diagram
- Next-state table

FSA and Languages

- Let A be an FSA with an input alphabet I. The set of all strings w from I* such that A goes to accepting state on w is called a language accepted by A: L(A)
- Eventual state-function N* : S I* S is a function that maps a pair {state, input string} to the state to which FSA would lead from the original state given the symbols in the input string as an input.

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 x1 and x2 from X if f(x1) = f(x2), then x1 = x2
- Determine whether the following functions are one-to-one:
- f : R R, f(x) = 4x – 1
- g : Z Z, g(n) = n2

- Hash functions

Onto Functions

- Function f : X Y is called onto (surjective) when given any element y from Y, there exists x in X so that f(x) = y
- Determine whether the following functions are onto:
- f : R R, f(x) = 4x – 1
- f : Z Z, g(n) = 4n – 1

- Bijection is one-to-one and onto
- Reversing strings function is bijective

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 cm,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, cm-1,n and cm-1,n-1
- Let F: Z Z Z and G: Z Z Z, F(n, m) = 3n6m and G(n, m) = 3n5m. Is F one-to-one, is G one-to-one?

Pigeonhole Principle

- If n pigeons fly into m pigeonholes and n > m, then at least one hole must contain two or more pigeons
- A function from one finite set to a smaller finite set cannot be one-to-one
- In a group of 13 people must there be at least two who have birthday in the same month?
- A drawer contains 10 black and 10 white socks. How many socks need to be picked to ensure that a pair is found?
- Let A = {1, 2, 3, 4, 5, 6, 7, 8}. If 5 integers are selected must at least one pair have sum of 9?

Pigeonhole Principle

- There is no FSA that accepts the following language: L = {s = akbk, 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 a1, a2, …, an is a sequence of n integers none of which is divisible by n. Show that at least one difference ai – aj is divisible by n

Composition of Functions

- Let f : X Y and g : Y Z, let range of f be a subset of the domain of g. The we can define a composition of g o f : X Z
- Let f,g : Z Z, f(n) = n + 1, g(n) = n2. Find f o g and g o f.
- Composition with identity function
- Composition with an inverse function
- Composition of two one-to-one functions is one-to-one
- Composition of two onto functions is onto

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

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