Definition:Logical Matrix

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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 $\set {1, 2, 3, 4}$ consists of the set of pairs:

$\set {\tuple {1, 1}, \tuple {1, 2}, \tuple {1, 3}, \tuple {1, 4}, \tuple {2, 2}, \tuple {2, 4}, \tuple {3, 3}, \tuple {4, 4} }$


This relation can be represented by the following logical matrix:

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