The problem presented involves the equation:

$|x - 1| = 0, \quad \forall x \in (0, 1)$

Let's break this down:

1. Absolute Value Property: The absolute value equation $|x - 1| = 0$ implies that the expression inside the absolute value must be zero. That is: $x - 1 = 0 \implies x = 1$
2. Domain Restriction: The condition $\forall x \in (0, 1)$ means that $x$ is restricted to values between 0 and 1, excluding 1.

Analysis:

• The equation $|x - 1| = 0$ is only true when $x = 1$.
• However, $x = 1$ is not in the interval $(0, 1)$, as the interval is open and excludes the endpoint 1.

Therefore, the equation $|x - 1| = 0$ cannot be true for any $x$ in $(0, 1)$, making the statement false.

Conclusion:

The correct choice is "False."

Would you like any further explanation or details? Here are some related questions to deepen understanding:

1. What is the definition of an open interval in set theory?
2. How would the answer change if the interval were closed, $[0, 1]$?
3. What are the general properties of absolute value equations?
4. How does the equation $|x - 1| = 0$ behave over different intervals?
5. Can we find any other intervals where $|x - 1| = 0$ holds?

Tip: Remember, absolute value equations are useful for finding distances from a reference point!