离散数学——Cryptography, Counting, Relation

目录

• Cryptography
  • RSA
• Counting
  • Generalized Pigeon Hole Theorem
  • The Bijection Theorem
  • Inclusion-Exclusion Principle
  • Pascal Identity
  • Binomial Theorem
  • Generating function
• Relation
  • Binary relation
  • Closure
  • Equivalence class
  • Poset
    • Hasse Diagram

Cryptography

RSA

The steps are

It correctness is proved by

Counting

Generalized Pigeon Hole Theorem

The Bijection Theorem

Inclusion-Exclusion Principle

Pascal Identity

Binomial Theorem

Generating function

A sequence {\(a_n\)} is like \(a_0, a_1, ... , a_n\)

The generating function of {\(a_n\)} is \(G(x) = a_0 + a_1x + a_2x + ... + a_nx^{n}\)

It has the following properties

We can use generating function in counting problem like

Relation

Binary relation

Binary relation from A to B is a subset of Cartesian product of A and B

A relation on A is relation from A to A

we can classify relation on one set to the followings

1. reflexive

The number of reflexive relations on a set A with \(|A| = n\) is \(2^{n(n−1)}\)

2. irreflexive

3. symmetric

4. antisymmetric

5. transitive

6. equivalence relation = reflexive + symmetric + transitive

7. partial ordering = reflexive + antisymmetric + transitive

We can define composite of relation, and also define power of relation

We can use this theorem to tell if a relation if transitive

\(R^n\) will link every "vertex" that has a "path" between them

Closure

The closure of a relation R with respect to property P is the minimum set S, that \(R \subseteq S\), and S has the property P

The transitive closure of a relation R is \(R*\), called the connectivity relation

Equivalence class

Poset

partial ordering = reflexive + antisymmetric + transitive

a set with a partial ordering R is called a poset

In a set S with a partial ordering R, a and b are two elements in S. If aRb or bRa exists, then we say a and b are comparable, otherwise incomparable

If every two elements in S are comparable, we say S is totally ordered

Hasse Diagram

Hasse Diagram: a diagram to represent a poset, but remove all the edges that must be present because of reflexive relation and transitive relation

Definition of well-ordered set

Definition of lattice