Answer to Question #273978 in Discrete Mathematics for Jayesh

Question

Represent each of these relations on {1, 2, 3} with a matrix (with the elements

of this set listed in increasing order).

a) {(1, 1), (1, 2), (1, 3)}

b) {(1, 2), (2, 1), (2, 2), (3, 3)}

c) {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)}

d) {(1, 3), (3, 1)}

Answer 1

Let us represent each of these relations on $\{1, 2, 3\}$ with a matrix (with the elements of this set listed in increasing order).

a) For the relation $\{(1, 1), (1, 2), (1, 3)\}$ the matrix is the following:

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

b) $\{(1, 2), (2, 1), (2, 2), (3, 3)\}$

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

c) $\{(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)\}$

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

d) $\{(1, 3), (3, 1)\}$

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