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Determine the elements of the set:

[tex]\[ M = \left\{\left(\frac{x^2-1}{2}\right) \in \mathbb{Z} \mid x \in \mathbb{Z}, -2 \leq x \leq 9\right\} \][/tex]

Sagot :

GinnyAnswer
Claro, procedamos a determinar los elementos del conjunto [tex]\(M\)[/tex].

Paso 1: Definir el rango de [tex]\(x\)[/tex]

El rango de [tex]\(x\)[/tex] se define entre [tex]\(A-2\)[/tex] y 9, donde [tex]\(A\)[/tex] es un valor dado. Supongamos que [tex]\(A = 2\)[/tex].

Por lo tanto, el rango de [tex]\(x\)[/tex] es: [tex]\(A-2 \leq x \leq 9\)[/tex]
[tex]\[2 - 2 \leq x \leq 9\][/tex]
[tex]\[0 \leq x \leq 9\][/tex]

Entonces, [tex]\(x\)[/tex] toma los valores: 0, 1, 2, 3, 4, 5, 6, 7, 8, y 9.

Paso 2: Aplicar la expresión [tex]\(\left( \frac{x^2 - 1}{2} \right)\)[/tex] para cada valor de [tex]\(x\)[/tex]

Vamos a evaluar la expresión para cada [tex]\(x\)[/tex]:

1. Para [tex]\(x = 0\)[/tex]:
[tex]\[\left( \frac{0^2 - 1}{2} \right) = \left( \frac{-1}{2} \right) = -1\][/tex]

2. Para [tex]\(x = 1\)[/tex]:
[tex]\[\left( \frac{1^2 - 1}{2} \right) = \left( \frac{0}{2} \right) = 0\][/tex]

3. Para [tex]\(x = 2\)[/tex]:
[tex]\[\left( \frac{2^2 - 1}{2} \right) = \left( \frac{4 - 1}{2} \right) = \left( \frac{3}{2} \right) = 1\][/tex]

4. Para [tex]\(x = 3\)[/tex]:
[tex]\[\left( \frac{3^2 - 1}{2} \right) = \left( \frac{9 - 1}{2} \right) = \left( \frac{8}{2} \right) = 4\][/tex]

5. Para [tex]\(x = 4\)[/tex]:
[tex]\[\left( \frac{4^2 - 1}{2} \right) = \left( \frac{16 - 1}{2} \right) = \left( \frac{15}{2} \right) = 7\][/tex]

6. Para [tex]\(x = 5\)[/tex]:
[tex]\[\left( \frac{5^2 - 1}{2} \right) = \left( \frac{25 - 1}{2} \right) = \left( \frac{24}{2} \right) = 12\][/tex]

7. Para [tex]\(x = 6\)[/tex]:
[tex]\[\left( \frac{6^2 - 1}{2} \right) = \left( \frac{36 - 1}{2} \right) = \left( \frac{35}{2} \right) = 17\][/tex]

8. Para [tex]\(x = 7\)[/tex]:
[tex]\[\left( \frac{7^2 - 1}{2} \right) = \left( \frac{49 - 1}{2} \right) = \left( \frac{48}{2} \right) = 24\][/tex]

9. Para [tex]\(x = 8\)[/tex]:
[tex]\[\left( \frac{8^2 - 1}{2} \right) = \left( \frac{64 - 1}{2} \right) = \left( \frac{63}{2} \right) = 31\][/tex]

10. Para [tex]\(x = 9\)[/tex]:
[tex]\[\left( \frac{9^2 - 1}{2} \right) = \left( \frac{81 - 1}{2} \right) = \left( \frac{80}{2} \right) = 40\][/tex]

Conclusión:
Los elementos del conjunto [tex]\(M\)[/tex] son los siguientes:
[tex]\[ M = \{-1, 0, 1, 4, 7, 12, 17, 24, 31, 40\} \][/tex]

Así, hemos determinado los elementos del conjunto [tex]\(M\)[/tex] de manera detallada y paso a paso.
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