Answer to Question #180701 in Discrete Mathematics for Durjoy

Question

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.

Answer 1

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.