# Definition:Logical Matrix

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## Definition

A **logical matrix** is a matrix whose entries are all either $0$ or $1$.

## Also known as

A **logical matrix** is also known as a **boolean matrix**, for George Boole.

## Examples

### Loop-Digraph

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

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

### 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 ordered 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**:

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

## Also see

- Results about
**logical matrices**can be found**here**.