Give the set in roster form.

{
x
2
−
9
∣
x
∈
Z
7
}

The given set in roster form represents the set of values obtained from the expression \(x^2 - 9\) where \(x\) belongs to the set of integers modulo 7 (\(Z_7\)), which is \(\{0, 1, 2, 3, 4, 5, 6\}\).

Let’s compute the values for each \(x\) from 0 to 6:

1. \(x = 0\): \(0^2 - 9 = 0 - 9 = -9\). Modulo 7, \(-9 \equiv -9 + 14 = 5 \mod 7\).
2. \(x = 1\): \(1^2 - 9 = 1 - 9 = -8\). Modulo 7, \(-8 \equiv -8 + 14 = 6 \mod 7\).
3. \(x = 2\): \(2^2 - 9 = 4 - 9 = -5\). Modulo 7, \(-5 \equiv -5 + 7 = 2 \mod 7\).
4. \(x = 3\): \(3^2 - 9 = 9 - 9 = 0\). Modulo 7, \(0 \equiv 0 \mod 7\).
5. \(x = 4\): \(4^2 - 9 = 16 - 9 = 7\). Modulo 7, \(7 \equiv 0 \mod 7\).
6. \(x = 5\): \(5^2 - 9 = 25 - 9 = 16\). Modulo 7, \(16 \equiv 2 \mod 7\).
7. \(x = 6\): \(6^2 - 9 = 36 - 9 = 27\). Modulo 7, \(27 \equiv 6 \mod 7\).

Thus, the set in roster form is:

\[
\{0, 2, 5, 6\}
\]

Would you like more details on how modulo operations work, or any clarifications?

Here are five related questions:
1. What is the modulus operation in number theory?
2. How do we compute squares in modular arithmetic?
3. Can the set of integers modulo \(n\) form a group?
4. What is the significance of quadratic residues in modular arithmetic?
5. How does modular arithmetic apply to cryptography?

**Tip:** When working with modulo operations, always ensure you bring negative results back into the positive range of the modulus!

Give the set in roster form for {x^2 - 9 | x ∈ Z_7}.

Find the set in roster form for x^2 - 9 with x belonging to Z_7 (integers modulo 7). This problem explores modular arithmetic, quadratic expressions, and the concept of quadratic residues. Step-by-step calculation for each value of x is provided.

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