My textbook, Microeconomic Theory by Mas-Colell, Whinston, and Green states that given a preference relation $\succsim$ on $X$,
Strict preference relation $\succ$ is defined by $ x \succ y \iff x \succsim y \text{ but not } y \succsim x $
Indifference relation $\sim$ is defined by $ x \sim y \iff x \succsim y \text{ and } y \succsim x $
Completeness property: For all $x,y \in X$, we have either $x \succsim y$ or $y \succsim x$ or both.
Suppose $\succsim$ on $X$ is complete. Define $A$ to be the truth set of $x \succsim y$, $B$ to be the truth set of $y \succsim x$.
Since $\succsim$ is complete, $A \cup B$ is the universal set.
First, $x \succ y \iff x \succsim y \land \lnot (y \succsim x)$, hence the truth set of $x \succ y$ is $A \setminus B$.
Next, $y \succ x \iff y \succsim x \land \lnot (x \succsim y)$, hence the truth set of $y \succ x$ is $B \setminus A$.
Finally, $x \sim y \iff x \succsim y \land y \succsim x$, hence the truth set of $x \sim y$ is $A \cap B$.
Using the identity $A \cup B = (A \setminus B) \sqcup (A \cap B) \sqcup (B \setminus A)$, is it safe to say that
If $\succsim$ is complete, then for any $x,y \in X$ exactly one of $x \succ y$ or $x \sim y$ or $y \succ x$ holds.
