计算理论导论(英文版)数学基础

15
N-ary relation

1-ary relation 2-ary relation 3-ary relation
unary relations binary relations ternary relations
16
17
5 composition
QR= {(a,b): for some c, (a,c)Q and (c,b) R }

Composition of f: AB and g:B C is a function h from A to C such that h(a)=g(f(a)).
18
6 special types of binary relations

directed graph


Node Edges Notes: do not allow “parallel arrows”.
2
written

Listing Refer to other sets and properties
3
4
5
Set operations

Union: AB Intersection: AB Complement: A Difference: A-B Disjoint Sets: AB=Null
29


A path in a binary relation R is a sequence (a1,..,an) for some n1 such that (ai, ai+1)R for i=1, …, n-1; this path is said to be from a1 to an. The length of a path (a1,..,an) is n. The path (a1,..,an) is a cycle if the ai’s are all distinct and also (an, a1) R .
38
Finite and Infinite Sets

1.4


Size We call two sets A and B equinumerous if there is a bijection f: A B . Equinumerosity is a symmetric relation Equinumerosity is a equivalence relation


For the objects belong to the relation No the objects not distinguished
11
Ordered Pairs
12
Cartesian product
13
Binary relation
14
Ordered tuple

Ordered 2-tuple
All of the elements on the diagonal line of its matrix are 1.
21
Symmetric


A relation RA×A is symmetric if (b,a) R whenever (a,b) R. In the directed graph representing a symmetric relation, whenever there is an arrow between two nodes, there are arrows between those nodes in both directions. Characteristics of matrix?
elements Color(R1) Shape(R2) Size(R3) x1 x2 x3 x4 x5 x6 x7 x8 red blue red blue yellow yellow red yellow round square triangular triangular round Square triangular triangular Small Large Small Small Small Small Large Large
Example: {(a,b): a and b are persons with the same father}
22
undirected graph

A symmetric relation without pairs of the form (a,a) is represented as an undirected graph, or simple a graph.
6
Laws and Properties


Commutative law Associative law Distributive law Absorption law Demogan’s law Idempotent law
7
Super Set

A collection of any sets

ordered pairs
Ordered Ordered Ordered Ordered


3-tuple ordered triples 4-tuple quadruples 5-tuple quintuples 6-tuple sextuples Sequence is an ordered n-tuple for some unspecified n, where n is the length of the sequence. N-fold Cartesian product
Chapter 0: Introduction, mathematical notation, proof techniques
1
0.2.1 Standard Set Theory review

A set is an unordered collection of objects.

Elements ( members) Multiset - Do distinguish repetitions of the elements {7,7} and {7}
R1={(x,y):x C, y S, and x is a city in state y} R2={(x, y):x S, y C, and y is a city in state x}
35
2 expression
Domain f(a) image of a under f


f: AB f(a) = b
1 0 if ( a , b ) f otherwise
19

Adjacent matrix
M [a, b] {
Properties of relations

Reflexive Symmetric

Asymmetric Anti-symmetric

Transitive
20
Reflexive
Argument
value
36
3 Certain kinds of functions

One-to-one 一对一 Onto 满射 Bijection 双射
37
4 Inverse of a binary relation R-1

Note



The inverse of a function need not be a function. If f is a bijection, f -1 is a function. f -1(f(a))=a , for each aA; f (f -1(b))=b , for each bB;a,b∈A ∧ a≠b源自∧ (a,b)R → (b,a)R
Example1: Let P be the set of all persons, {(a,b): a ,b P and a is the father of b}
25
Transitive


A relation RA×A is transitive if whenever (a , b) R and (b, c) R, then (a , c) R. In terms of the directed graph representation, transitivity is equivalent to the requirement that whenever there is a sequence of arrows leading from an element a to an element z, there is an arrow directly from a to z.
23
Asymmetric (非对称性）

A relation RA×A is asymmetric if： ∃(a, b ∈ A ∧ (a , b) R → (b , a) R )
24
Antisymmetric (反对称性)

A relation RA×A is anti-symmetric :
40
infinite
S S
8
9
Coverage

Difference between coverage and partition
10
0.2.2 sequences and tuples


How to express relations between objects The language of sets