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.