Draw the digraph and the matrix of the relation R= {(1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3,4), (4, 1), (4, 2), (4, 3)} on the set A= {1, 2, 3, 4, 5}. Also decide whether it is reflexive, whether it is symmetric, whether it is anti symmetric,whether it is transitive.

Above is the digraph for the relation

The relation is not refexive. This is because according to definition of reflexive, $aRa \forall a\in A$ but $3\not R 3, 4 \not R4$ etc. Hence the relation is not reflexive.

Also, the relation is not symmetric. By definition of symmetric, if $aRb$ ,then $bRa \forall a,b \in A$ . But, $2R3$ and $3 \not R 2.$

Hence, the relation is not symmetric.

For anti-symmetry, if $aRb$ and $bRa \implies a=b \forall a,b \in A$ . 3R4 and 4R3 but, $3 \neq 4.$ Hence the relation is not anti-symmetric.

For transitive, if $aRb$ and $bRc,$ then $aRc \forall a,b,c\in A$ . 1R3 and 3R4 ,but $1 \not R 4$ . Hence the relation is not transitive.