Discrete Math Notes:
Chapter 5: Functions and Relations
5.11 Matrix multiplication and graph powers
Matrix, a set S is an array of elements from S with n rows and m columns.
Entry, Each element in a matrix
square matrix, the number of rows is equal to the number of columns
adjacency matrix, directed graph G with n vertices can be represented by an n n matrix over the set {0, 1}
Boolean matrix, a matrix whose entries are from the set {0, 1}.
dot product,
matrix product, If A and B are n x n matrices over the integers,
power of a matrix, the product of k copies of A
matrix sum, If A and B are two m x n matrices
overview:
An array of elements from S with n rows and m columns is called a n x m matrix over a set S. An entry is the name given to each element in a matrix. If the number of rows equals the number of columns, the
adding and multiplying items in a Boolean matrix, Boolean addition and multiplication are utilized. Because those operations can be used to compute the transitive closure of a graph, this material covers matrix addition and multiplication for square Boolean matrices.
5.12 Partial orders
partial order, relation R on a set A can be partial order if it is reflexive, transitive, and anti-symmetric
partially ordered set or poset, The domain along with a partial order defined on it is denoted (A, )
comparable, if x y or y x.
Hasse diagram, named after the 20th century German mathematician Helmut Hasse, is a useful way to depict a partial order on a finite set.
Overview:
A={a,b,c,d,e,f} Partial order Pon A
P is a partial order: P is reflexive p is anj
No:
P is transitive
(a,c)and(c,e)>(ae) The minimal elernents are a and d: There are no arrows into a or d except the ones from themselves
The maximal elements are e and f: The only edges leaving e and f point to thermselves
Captions ^
1. P is reflexive (all self-loops are present) 2. P is anti-symmetric (there are no two elements that point to each other) 3. P is transitive (edges (a,c) and (c,e) imply the presence of edge (a,e)). Therefore, P is a partial order. 4. a and d are minimal elements because there are no arrows into a or d except the ones from themselves. 5. e and f are maximal elements because the only edges leaving e and f point to themselves.
If a relation R on a set A is reflexive, transitive, and anti-symmetric, it is a partial order. aRb is written with the notation a b, which reflects the fact that a partial order acts as an operator on the elements of A. "A is at most b," reads the sentence a b. A partially ordered set or poset is defined by the domain and a partial order defined on it, and is denoted by (A,). I