Let's analyze the given problem.

We are dealing with an equivalence relation defined by modulo 7 on the set of integers, denoted as $\mathbb{Z}$. For any integer $x$, the notation $[x]$ represents the equivalence class of $x$ under this relation, meaning all integers that are congruent to $x \mod 7$.

We need to determine which of the given options is not correct. Let's check each one.

Step-by-step evaluation of each option:

(a) $[2] = [-5]$

• $2 \equiv -5 \pmod{7}$?
• $-5 + 7 = 2$, so yes, $2 \equiv -5 \pmod{7}$.

This option is correct.

(b) $[0] = [7]$

• $0 \equiv 7 \pmod{7}$?
• $7 \mod 7 = 0$, so yes, $0 \equiv 7 \pmod{7}$.

This option is correct.

(c) $[6] \neq [-1]$

• $6 \equiv -1 \pmod{7}$?
• $-1 + 7 = 6$, so $6 \equiv -1 \pmod{7}$. Hence, $[6] = [-1]$.

This option is incorrect**, since they are congruent.**

(d) $[3] = [-4]$

• $3 \equiv -4 \pmod{7}$?
• $-4 + 7 = 3$, so yes, $3 \equiv -4 \pmod{7}$.

This option is correct.

Conclusion:

The incorrect option is (c) $[6] \neq [-1]$, because $[6] = [-1]$.

Would you like further details on this solution or explanations about equivalence relations?

Here are 5 related questions:

1. What is the general definition of an equivalence relation?
2. How do equivalence classes behave under modular arithmetic?
3. How can we verify congruence between two integers modulo $n$?
4. Can an equivalence class under $\mod 7$ contain negative integers?
5. How does modular arithmetic simplify number theory problems?

Tip: When working with congruences, always simplify negative numbers by adding the modulus to bring them into the standard range.