Finding the matrix of a relation $R$ on $X$

Question

Given the relation:

$$ R = \{(x, y)|x < y\} $$ and the ordering of $x$ is $\{ 1,2,3 \}.$
How to find the matrix of this relation?

I honestly cannot understand the question. Any tips to solve this problem?

DISC 101, Relations

Answer 1

If $\mathcal{R}$ is a relation between $X$ and $Y$ and $x_1, ..., x_m$ is an ordering of the elements of $X$ and $y_1, ..., y_n$ is an ordering of the elements of $Y$, then the matrix $M$ of $\mathcal{R}$ is a $m \times n$ matrix such that $m_{i,j} = 1$ if $x_i \mathcal{R} y_j$ and $0$ otherwise.

In your example, $X = Y = \{1,2,3\}$ and the ordering is $1, 2, 3$.
$M$ is therefore a $3 \times 3$ matrix, such that $m_{i,j} = 1$ if $i < j$ and $0$ otherwise. $$M = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$$