Which one of the following relations on R is an equivalence relation?

The correct option is aR₁b ⇔ |a| = |b|.

An equivalence relation on a set is a relation that satisfies three fundamental properties: reflexivity, symmetry, and transitivity. Let us verify these three properties for the relation $R_1$ defined on the set of real numbers $ℝ$ by $a{R}_{1}b\iff \left|a\right|=\left|b\right|$.

1. Reflexivity:
A relation $R$ is reflexive if every element is related to itself, i.e., $aRa$ for all $a\in ℝ$.
For any real number $a$, it is always true that:
$\left|a\right|=\left|a\right|$
Thus, $a{R}_{1}a$ holds true. The relation is reflexive.

2. Symmetry:
A relation $R$ is symmetric if $aRb$ implies $bRa$ for all $a,b\in ℝ$.
If $a{R}_{1}b$, then by definition:
$\left|a\right|=\left|b\right|$
Since equality is symmetric, this directly implies:
$\left|b\right|=\left|a\right|$
Hence, $b{R}_{1}a$ holds true. The relation is symmetric.

3. Transitivity:
A relation $R$ is transitive if $aRb$ and $bRc$ imply $aRc$ for all $a,b,c\in ℝ$.
Assume $a{R}_{1}b$ and $b{R}_{1}c$. This means:
$\left|a\right|=\left|b\right|$ and $\left|b\right|=\left|c\right|$
By the transitive property of equality, we get:
$\left|a\right|=\left|c\right|$
Therefore, $a{R}_{1}c$ holds true. The relation is transitive.

Since the relation $R_1$ is reflexive, symmetric, and transitive, it is an equivalence relation on $ℝ$.