Considere o \mathbb{Z}_{2} a tabela abaixo mostra todas as possibilidades da soma e produto dos seus elementos:
+ \overline{0} \overline{1}
\overline{0} \overline{0} \overline{1}
\overline{1} \overline{1} \overline{0}
\cdot \overline{0} \overline{1}
\overline{0} \overline{0} \overline{0}
\overline{1} \overline{1} \overline{0}
Faça as tabelas das operações para os conjuntos \mathbb{Z}_{3}, \mathbb{Z}_{4}, \mathbb{Z}_{5}, \mathbb{Z}_{6}, \mathbb{Z}_{7}, \mathbb{Z}_{8} e \mathbb{Z}_{9}.

Passo 1

O enunciado pede para construir, para cada inteiro n=3,4,5,6,7,8,9, as tabelas de adição e multiplicação no anel \mathbb{Z}_n. O conteúdo é aritmética modular: dado a,b\in\{0,1,\dots,n-1\}, definimos
a + b \equiv (a+b)\bmod n, \qquad a \cdot b \equiv (a\cdot b)\bmod n.

Passo 2

O método geral para montar cada tabela é:
  1. Listar na primeira linha (cabeçalho) os elementos \overline{0},\overline{1},\dots,\overline{n-1}.
  2. Para cada linha rotulada por \overline{i}, e cada coluna \overline{j}, calcular \overline{i+j} ou \overline{i\cdot j} em módulo n.
  3. Preencher a célula correspondente com \overline{k}, onde k\equiv i+j\pmod n ou k\equiv i\cdot j\pmod n.
A seguir, aplicamos isso para cada \mathbb{Z}_n.

Passo 3

Tabelas em \mathbb{Z}_{3}:
[
+ \overline{0} \overline{1} \overline{2}
\overline{0} \overline{0} \overline{1} \overline{2}
\overline{1} \overline{1} \overline{2} \overline{0}
\overline{2} \overline{2} \overline{0} \overline{1}
]
[
\cdot \overline{0} \overline{1} \overline{2}
\overline{0} \overline{0} \overline{0} \overline{0}
\overline{1} \overline{0} \overline{1} \overline{2}
\overline{2} \overline{0} \overline{2} \overline{1}
]

Passo 4

Tabelas em \mathbb{Z}_{4}:
[
+ \overline{0} \overline{1} \overline{2} \overline{3}
\overline{0} \overline{0} \overline{1} \overline{2} \overline{3}
\overline{1} \overline{1} \overline{2} \overline{3} \overline{0}
\overline{2} \overline{2} \overline{3} \overline{0} \overline{1}
\overline{3} \overline{3} \overline{0} \overline{1} \overline{2}
]
[
\cdot \overline{0} \overline{1} \overline{2} \overline{3}
\overline{0} \overline{0} \overline{0} \overline{0} \overline{0}
\overline{1} \overline{0} \overline{1} \overline{2} \overline{3}
\overline{2} \overline{0} \overline{2} \overline{0} \overline{2}
\overline{3} \overline{0} \overline{3} \overline{2} \overline{1}
]

Passo 5

Tabelas em \mathbb{Z}_{5}:
[
+ \overline{0} \overline{1} \overline{2} \overline{3} \overline{4}
\overline{0} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4}
\overline{1} \overline{1} \overline{2} \overline{3} \overline{4} \overline{0}
\overline{2} \overline{2} \overline{3} \overline{4} \overline{0} \overline{1}
\overline{3} \overline{3} \overline{4} \overline{0} \overline{1} \overline{2}
\overline{4} \overline{4} \overline{0} \overline{1} \overline{2} \overline{3}
]
[
\cdot \overline{0} \overline{1} \overline{2} \overline{3} \overline{4}
\overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0}
\overline{1} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4}
\overline{2} \overline{0} \overline{2} \overline{4} \overline{1} \overline{3}
\overline{3} \overline{0} \overline{3} \overline{1} \overline{4} \overline{2}
\overline{4} \overline{0} \overline{4} \overline{3} \overline{2} \overline{1}
]

Passo 6

Tabelas em \mathbb{Z}_{6}:
[
+ \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5}
\overline{0} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5}
\overline{1} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{0}
\overline{2} \overline{2} \overline{3} \overline{4} \overline{5} \overline{0} \overline{1}
\overline{3} \overline{3} \overline{4} \overline{5} \overline{0} \overline{1} \overline{2}
\overline{4} \overline{4} \overline{5} \overline{0} \overline{1} \overline{2} \overline{3}
\overline{5} \overline{5} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4}
]
[
\cdot \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5}
\overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0}
\overline{1} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5}
\overline{2} \overline{0} \overline{2} \overline{4} \overline{0} \overline{2} \overline{4}
\overline{3} \overline{0} \overline{3} \overline{0} \overline{3} \overline{0} \overline{3}
\overline{4} \overline{0} \overline{4} \overline{2} \overline{0} \overline{4} \overline{2}
\overline{5} \overline{0} \overline{5} \overline{4} \overline{3} \overline{2} \overline{1}
]

Passo 7

Tabelas em \mathbb{Z}_{7}:
[
+ \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6}
\overline{0} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6}
\overline{1} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{0}
\overline{2} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{0} \overline{1}
\overline{3} \overline{3} \overline{4} \overline{5} \overline{6} \overline{0} \overline{1} \overline{2}
\overline{4} \overline{4} \overline{5} \overline{6} \overline{0} \overline{1} \overline{2} \overline{3}
\overline{5} \overline{5} \overline{6} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4}
\overline{6} \overline{6} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5}
]
[
\cdot \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6}
\overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0}
\overline{1} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6}
\overline{2} \overline{0} \overline{2} \overline{4} \overline{6} \overline{1} \overline{3} \overline{5}
\overline{3} \overline{0} \overline{3} \overline{6} \overline{2} \overline{5} \overline{1} \overline{4}
\overline{4} \overline{0} \overline{4} \overline{1} \overline{5} \overline{2} \overline{6} \overline{3}
\overline{5} \overline{0} \overline{5} \overline{3} \overline{1} \overline{6} \overline{4} \overline{2}
\overline{6} \overline{0} \overline{6} \overline{5} \overline{4} \overline{3} \overline{2} \overline{1}
]

Passo 8

Tabelas em \mathbb{Z}_{8}:
[
+ \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7}
\overline{0} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7}
\overline{1} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{0}
\overline{2} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{0} \overline{1}
\overline{3} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{0} \overline{1} \overline{2}
\overline{4} \overline{4} \overline{5} \overline{6} \overline{7} \overline{0} \overline{1} \overline{2} \overline{3}
\overline{5} \overline{5} \overline{6} \overline{7} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4}
\overline{6} \overline{6} \overline{7} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5}
\overline{7} \overline{7} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6}
]
[
\cdot \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7}
\overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0}
\overline{1} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7}
\overline{2} \overline{0} \overline{2} \overline{4} \overline{6} \overline{0} \overline{2} \overline{4} \overline{6}
\overline{3} \overline{0} \overline{3} \overline{6} \overline{1} \overline{4} \overline{7} \overline{2} \overline{5}
\overline{4} \overline{0} \overline{4} \overline{0} \overline{4} \overline{0} \overline{4} \overline{0} \overline{4}
\overline{5} \overline{0} \overline{5} \overline{2} \overline{7} \overline{4} \overline{1} \overline{6} \overline{3}
\overline{6} \overline{0} \overline{6} \overline{4} \overline{2} \overline{0} \overline{6} \overline{4} \overline{2}
\overline{7} \overline{0} \overline{7} \overline{6} \overline{5} \overline{4} \overline{3} \overline{2} \overline{1}
]

Passo 9

Tabelas em \mathbb{Z}_{9}:
[
+ \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{8}
\overline{0} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{8}
\overline{1} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{8} \overline{0}
\overline{2} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{8} \overline{0} \overline{1}
\overline{3} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{8} \overline{0} \overline{1} \overline{2}
\overline{4} \overline{4} \overline{5} \overline{6} \overline{7} \overline{8} \overline{0} \overline{1} \overline{2} \overline{3}
\overline{5} \overline{5} \overline{6} \overline{7} \overline{8} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4}
\overline{6} \overline{6} \overline{7} \overline{8} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5}
\overline{7} \overline{7} \overline{8} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6}
\overline{8} \overline{8} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7}
]
[
\cdot \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{8}
\overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0} \overline{0}
\overline{1} \overline{0} \overline{1} \overline{2} \overline{3} \overline{4} \overline{5} \overline{6} \overline{7} \overline{8}
\overline{2} \overline{0} \overline{2} \overline{4} \overline{6} \overline{8} \overline{1} \overline{3} \overline{5} \overline{7}
\overline{3} \overline{0} \overline{3} \overline{6} \overline{0} \overline{3} \overline{6} \overline{0} \overline{3} \overline{6}
\overline{4} \overline{0} \overline{4} \overline{8} \overline{3} \overline{7} \overline{2} \overline{6} \overline{1} \overline{5}
\overline{5} \overline{0} \overline{5} \overline{1} \overline{6} \overline{2} \overline{7} \overline{3} \overline{8} \overline{4}
\overline{6} \overline{0} \overline{6} \overline{3} \overline{0} \overline{6} \overline{3} \overline{0} \overline{6} \overline{3}
\overline{7} \overline{0} \overline{7} \overline{5} \overline{3} \overline{1} \overline{8} \overline{6} \overline{4} \overline{2}
\overline{8} \overline{0} \overline{8} \overline{7} \overline{6} \overline{5} \overline{4} \overline{3} \overline{2} \overline{1}
]

Resposta

As tabelas acima são, para cada n=3,4,5,6,7,8,9, as operações de soma e produto em \mathbb{Z}_n.

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