Question

Draw directed graphs representing relations of the following types.

Answer 1

Answer:

The first graph in the picture describes a relation that is (a) Reflexive, transitive, and anti-symmetric.

Because a, b, c, d all are related to itself, so reflexive.

where (a, b) and (b, d) are in the relation, (a,d) is in the relation,    

for (c,a) and (a,b) there is (c,b).

so it's transitive.

for all a,b in the relation, (a,b) there is no (b,a) with a ≠b.

The second graph in the picture describes a relation that is (b) Reflexive, transitive, and neither symmetric nor anti-symmetric.

Because a, b, c, d all are related to itself, so reflexive.

where (a, b) and (b, a) are in the relation, (a,a) is in the relation,                                                    

where (c, d) and (d, d) are in the relation, (c,d) is in the relation,                                                    

so it's transitive.

Because, (a,b) and (b.a) are there, but for (c,d) there is no (d,c) in relation.

So, the relation is not symmetric.

(a,b) and (b,a) is in relation but, a≠b, so not anti symmetric.

Explanation:

For all a in a set,  if (a,a) in a relation then the relation is reflexive.

For all (a,b) in relation R, if (b,a) is also in R, then R is symmetric.

For all (a,b), (b,c) in relation R, if (a,c) is also in R, then R is transitive.

For all (a,b), (b,a) in R, a = b,  then R is an anti- symmetric relation.