For the equivalence relation equiv(mod 9 ) defined on Z, let [x] denote the equivalence relation of x. Find the correct option.

(a) [2] ne[11]

(b) [0] ne[9]

(c) [6] = [- 1]

(d) [3] = [- 6]

We are dealing with equivalence relations defined modulo \(9\), where two integers \(x\) and \(y\) are equivalent modulo \(9\) if their difference is divisible by \(9\). In other words:

\[
x \equiv y \pmod{9} \iff x - y \text{ is divisible by } 9
\]

Let’s analyze each option:

**(a) \([2] \neq [11]\)**

- \(2 \equiv 11 \pmod{9}\), since \(11 - 2 = 9\), which is divisible by \(9\). 
Thus, \( [2] = [11] \), so this option is false.

**(b) \([0] \neq [9]\)**

- \(0 \equiv 9 \pmod{9}\), since \(9 - 0 = 9\), which is divisible by \(9\). 
Thus, \( [0] = [9] \), so this option is false.

**(c) \([6] = [-1]\)**

- \(6 \equiv -1 \pmod{9}\), since \(6 - (-1) = 7\), which is **not divisible** by \(9\).
Thus, \( [6] \neq [-1] \), so this option is false.

**(d) \([3] = [-6]\)**

- \(3 \equiv -6 \pmod{9}\), since \(3 - (-6) = 9\), which is divisible by \(9\).
Thus, \( [3] = [-6] \), so this option is true.

### **Correct option: (d)**

Would you like any further clarification or details on this? 

Here are 5 related questions to consider:
1. What is the general rule for equivalence modulo \(n\)?
2. How do you compute the equivalence class of a number?
3. Can two negative numbers be equivalent modulo \(9\)?
4. Why is \( [6] \neq [-1] \) in this problem?
5. How can we represent all the equivalence classes for modulo \(9\)?

**Tip**: Always check equivalence by computing the difference and verifying if it's divisible by the modulus.

For the equivalence relation equiv(mod 9) defined on Z, let [x] denote the equivalence relation of x. Find the correct option.

(a) [2] ≠ [11]

(b) [0] ≠ [9]

(c) [6] = [-1]

(d) [3] = [-6]

Explore how to solve a problem involving equivalence relations modulo 9. This step-by-step solution covers modular arithmetic, equivalence classes, and how to check if two numbers are equivalent modulo 9. Ideal for high school students (Grades 10-12).

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