Vitali Set Existence Theorem/Lemma

Theorem

For all real numbers in the closed unit interval $\mathbb I = \closedint 0 1$, define the relation $\sim$ such that:

$\forall x, y \in \mathbb I: x \sim y \iff x - y \in \Q$

where $\Q$ is the set of rational numbers.

That is, $x \sim y$ if and only if their difference is rational.

Then $\sim$ is an equivalence relation.

Proof

Checking in turn each of the criteria for equivalence:

Reflexivity

$\forall x \in \mathbb I: x - x = 0 \in \Q$

Thus $\sim$ is reflexive.

$\Box$

Symmetry

By Inverse for Rational Addition:

$\forall x, y \in \mathbb I: x - y \in \Q \implies y - x = -\paren {x - y} \in \Q$

Thus $\sim$ is symmetric.

$\Box$

Transitivity

Let $x, y, z \in \mathbb I$.

Suppose $x - y \in \Q$ and $y - z \in \Q$.

Then by Rational Addition is Closed:

$x - z = \paren {x - y} + \paren {y - z} \in \Q$

Thus $\sim$ is transitive.

$\Box$

Thus we have shown that $\sim$ is an equivalence relation.

$\blacksquare$