Which of the following relations is transitive but not reflexive for the set S={3, 4, 6}?

The correct option is: R = {(3, 4), (4, 6), (3, 6)}

To understand why this relation is transitive but not reflexive for the set
$S=\{3,4,6\}$
we must analyze the definitions of reflexive and transitive relations.

1. Reflexive Relation:
A relation $R$ on a set $S$ is reflexive if every element in the set is related to itself. In other words, for every element
$a\in S$
the ordered pair
$(a,a)$
must belong to $R$.

For our set
$S=\{3,4,6\}$
to be reflexive, the relation $R$ must contain all of the following pairs:
$(3,3),(4,4), \text{and} (6,6)$
Checking our relation
$R=\left\{\right(3,4),(4,6),(3,6\left)\right\}$
we observe that none of these reflexive pairs are present. Thus, the relation $R$ is not reflexive.

2. Transitive Relation:
A relation $R$ is transitive if whenever
$(a,b)\in R$
and
$(b,c)\in R$
then the pair
$(a,c)$
must also be in $R$.

Let us check the elements in
$R=\left\{\right(3,4),(4,6),(3,6\left)\right\}$
- We have the pair
$(3,4)\in R$
- We have the pair
$(4,6)\in R$
- For transitivity, since 4 is the connecting element, the pair
$(3,6)$
must exist in $R$. Looking at the definition of $R$, we see that
$(3,6)$
is indeed present.

Since there are no other pairs in $R$ that can be combined in this manner, the transitivity condition is satisfied. Hence, the relation is transitive.

Therefore, the relation
$R=\left\{\right(3,4),(4,6),(3,6\left)\right\}$
is transitive but not reflexive.