Before we discuss the properties of binary relations in general, we first introduce some ways to represent a given binary relation.# Explanation: Representations of Binary Relations

(related to Chapter: Binary Relations and Their Properties)

There are different possibilities to represent a binary relation $R\subseteq S\times T$. Sometimes, we can just enumerate the elements of $$R$$. For instance, if $$S,T$$ are bothe the set of natural numbers $$\mathbb N$$ and $$R$$ is the successor relation, then we can write $R:=\{(1,2),(2,3),(3,4),\ldots \}$

The figure below shows another representation of an infinite relation of all pairs of real numbers $$x,y\in\mathbb R$$ satisfying the relation $$x\le y$$. If $$S, T$$ have a finite number of elements, there are at least four possibilities to represent a relation. We demonstrate these possibilities for the divisibility relation $$R$$ of all natural numbers between 1 and 10.

#### 1) Table Representation of $$R$$:

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 :------------- |:------------- |:------------- |:------------- |:------------- |:------------- |:------------- |:------------- |:------------- |:------------- |:------------- 1| x| x| x| x| x| x| x| x| x| x 2| | x| | x| | x| | x| | x 3| | | x| | | x| | | x| 4| | | | x| | | | x| | 5| | | | | x| | | | | x 6| | | | | | x| | | | 7| | | | | | | x| | | 8| | | | | | | | x| | 9| | | | | | | | | x| 10| | | | | | | | | | x

#### 2) Matrix. Representation of $$R$$:

$M_R:=\left(\begin{array}{cccccccccc} 1&1&1&1&1&1&1&1&1&1\\ 0&1&0&1&0&1&0&1&0&1\\ 0&0&1&0&0&1&0&0&1&0\\ 0&0&0&1&0&0&0&1&0&0\\ 0&0&0&0&1&0&0&0&0&1\\ 0&0&0&0&0&1&0&0&0&0\\ 0&0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&0&1\\ \end{array} \right)$

Please note that the matrix of the inverse relation $R^{-1}$ corresponds to the transposed matrix of $M_R$. i.e.

$M_{R^{-1}}=(M_R)^{T}=\left(\begin{array}{cccccccccc} 1&0&0&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0&0&0\\ 1&0&1&0&0&0&0&0&0&0\\ 1&1&0&1&0&0&0&0&0&0\\ 1&0&0&0&1&0&0&0&0&1\\ 1&1&1&0&0&1&0&0&0&0\\ 1&0&0&0&0&0&1&0&0&0\\ 1&1&0&1&0&0&0&1&0&0\\ 1&0&1&0&0&0&0&0&1&0\\ 1&1&0&0&0&0&0&0&0&1\\ \end{array} \right)$ In our case, this matrix represents the relation of numbers $(n,m)$ where $n$ is the multiple of $m.$

#### 3) Digraph. Representation of $$R$$: The digraph of the inverse relation is the same digraph with all arrows reversed.

#### 4) List Representation of $$R$$:

is devisor of ... 1| $$\{1,2,3,4,5,6,7,8,9,10\}$$ 2| $$\{2,4,6,8,10\}$$ 3| $$\{3,6,9\}$$ 4| $$\{4,8\}$$ 5| $$\{5,10\}$$ 6| $$\{6\}$$ 7| $$\{7\}$$ 8| $$\{8\}$$ 9| $$\{9\}$$ 10| $$\{10\}$$

Explanations: 1

Github: ### References

#### Bibliography

1. Schmidt G., Ströhlein T.: "Relationen und Graphen", Springer-Verlag, 1989