Question

Write the addition and multiplication tables for \(\mathbb{Z}_{4}\).

Short answer

The addition table for \(\mathbb{Z}_{4}\) is: \(\begin{array}{c|c c c c} + & 0 & 1 & 2 & 3 \ \hline 0 & 0 & 1 & 2 & 3 \ 1 & 1 & 2 & 3 & 0 \ 2 & 2 & 3 & 0 & 1 \ 3 & 3 & 0 & 1 & 2 \end{array}\). The multiplication table for \(\mathbb{Z}_{4}\) is: \(\begin{array}{c|c c c c} * & 0 & 1 & 2 & 3 \ \hline 0 & 0 & 0 & 0 & 0 \ 1 & 0 & 1 & 2 & 3 \ 2 & 0 & 2 & 0 & 2 \ 3 & 0 & 3 & 2 & 1 \end{array}\).

Step-by-step solution

Set Up the Tables

Begin by setting up the tables. Each table should be 4x4 as there are four elements in \(\mathbb{Z}_{4}\): 0, 1, 2, and 3. Label both the rows and columns of the table with these numbers. Do this twice, one for the addition table and one for the multiplication table.

Create Addition Table

To create the addition table, start with the upper-left corner. The top row and the left column should be the numbers of \(\mathbb{Z}_4\) in order. Add the row number with the column number modulo 4. Fill out each cell of the table based on this sum modulo 4. For example, the cell in the '3' row and '2' column would contain the number '1', because (3+2) modulo 4 is 1.

Create Multiplication Table

Similar to the addition table, the multiplication table is created by multiplying the row number with the column number modulo 4. Fill out each cell based on this product. For example, the cell in the '3' row and '2' column would contain the number '2', because (3*2) modulo 4 is 2.