Definition:Logical Matrix

From ProofWiki
Jump to navigation Jump to search

Definition

A logical matrix (or boolean matrix) is a matrix whose entries are all either $0$ or $1$.


Examples

Loop-Digraph

The adjacency matrix for a loop-digraph is a logical matrix:

LoopDigraph.png $\qquad \begin{pmatrix} 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ \end{pmatrix}$


Relation

A logical matrix can be used as an explicit definition of a binary relation.

The binary relation is a divisor of on the set of natural numbers $\left\{{1, 2, 3, 4}\right\}$ consists of the set of pairs:

$\left\{{(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}\right\}$


This relation can be represented by the following logical matrix:

$\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$