What type of relation is ‘less than’ in the set of real numbers?

The correct option is "only transitive".

To understand why the "less than" relation ($<$) on the set of real numbers ($ℝ$) is only transitive, let us analyze the three standard properties of relations: reflexivity, symmetry, and transitivity.

1. Reflexivity:
A relation $R$ on a set is reflexive if every element is related to itself. For the "less than" relation, this would mean for any real number $a$, we must have:
$a<a$
However, no real number is strictly less than itself (for example, $5<5$ is false). Therefore, the relation is not reflexive.

2. Symmetry:
A relation is symmetric if $aRb$ implies $bRa$. For the "less than" relation, this means if:
$a<b$
then it must follow that:
$b<a$
But this is false. For example, since $2<3$, it is mathematically impossible for $3<2$ to be true. Therefore, the relation is not symmetric.

3. Transitivity:
A relation is transitive if $aRb$ and $bRc$ together imply $aRc$. For the "less than" relation, let $a$, $b$, and $c$ be any real numbers such that:
$a<b$ and $b<c$
By the basic order properties of real numbers, if $a$ is strictly smaller than $b$, and $b$ is strictly smaller than $c$, then $a$ must be strictly smaller than $c$:
$a<c$
Since this condition holds true for all real numbers, the relation is transitive.

Since the relation is transitive but neither reflexive nor symmetric, it is not an equivalence relation. Among the choices provided, the only accurate description is that it is only transitive.