https://file.aiquickdraw.com/mathbot/file/1727250037444e5gwdsc5.jpg

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 \( [6] = [-1] \).

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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.

For the equivalence relation ≡ (mod 7) defined on ℤ, let [x] denote the equivalence relation of x. Find the option which is not correct:
(a) [2] = [-5]
(b) [0] = [7]
(c) [6] ≠ [-1]
(d) [3] = [-4]

Solve the modular arithmetic problem involving equivalence relations under mod 7. Understand the equivalence class and congruence relations, and find the incorrect option. Suitable for Grades 10-12.

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