To determine which numbers belong to the equivalence class $[9]$ modulo 11, we need to understand the concept of modular equivalence.

Definition

The equivalence class $[9]$ modulo 11 consists of all integers $x$ such that:
$x \equiv 9 \pmod{11}$ This means that the numbers in this equivalence class differ from 9 by a multiple of 11.

General Form

A number belongs to $[9]$ modulo 11 if it can be written as: $x = 9 + 11k, \quad \text{for any integer } k \in \mathbb{Z}.$

Examples

For different values of $k$, we get:

• $k = 0 \Rightarrow 9$
• $k = 1 \Rightarrow 9 + 11(1) = 20$
• $k = -1 \Rightarrow 9 + 11(-1) = -2$
• $k = 2 \Rightarrow 9 + 11(2) = 31$
• $k = -2 \Rightarrow 9 + 11(-2) = -13$

Thus, the equivalence class $[9]$ modulo 11 contains numbers such as: $\{..., -13, -2, 9, 20, 31, ...\}$ which follow the pattern $x = 9 + 11k$.

Conclusion

Any number of the form $x = 9 + 11k$, where $k$ is any integer, belongs to the equivalence class $[9]$ modulo 11.

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Related Questions

1. What is an equivalence class in modular arithmetic?
2. How do you determine equivalence classes for different moduli?
3. What are all the equivalence classes modulo 11?
4. How can equivalence classes be used in number theory?
5. How do you determine if two numbers are in the same equivalence class?

Tip

In modular arithmetic, two numbers $a$ and $b$ are in the same equivalence class modulo $n$ if their difference is a multiple of $n$, i.e., $a - b$ is divisible by $n$.