Question

Write out the addition and multiplication tables for

a)${\mathrm{\mathbb{Z}}}_2$b)${\mathrm{\mathbb{Z}}}_4$c)${\mathrm{\mathbb{Z}}}_7$d)${\mathrm{\mathbb{Z}}}_{12}$

Short answer

"Thus, the required addition and multiplication tables are obtained for each part".

Step-by-step solution

Addition table for  n=2

Consider the given value.

for addition and multiplication, and drawing the table:

$\oplus$	[0]	[1]
[0]	[0]	[1]
[1]	[1]	[0]

Multiplication for table

$⊝$	[0]	[1]
[0]	[0]	[0]
[1]	[1]	[1]

Addition table for  n=4

Consider the given value

for addition and multiplication, and drawing the table:

$\oplus$	[0]	[1]	[2]	[3]
[0]	[0]	[1]	[2]	[3]
[1]	[1]	[2]	[3]	[0]
[2]	[2]	[3]	[0]	[1]
[3]	[3]	[0]	[1]	[2]

Multiplication for table $n=4$

$⊝$	[0]	[1]	[2]	[3]
[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]
[2]	[0]	[2]	[0]	[2]
[3]	[0]	[3]	[2]	[1]

Addition table for  n=7

Consider the given value

for addition and multiplication, and drawing the table:

$\oplus$	[0]	[1]	[2]	[3]	[4]	[5]	[6]
[0]	[0]	[1]	[2]	[3]	[4]	[5]	[6]
[1]	[1]	[2]	[3]	[4]	[5]	[6]	[0]
[2]	[2]	[3]	[4]	[5]	[6]	[0]	[1]
[3]	[3]	[4]	[5]	[6]	[0]	[1]	[2]
[4]	[4]	[5]	[6]	[0]	[1]	[2]	[3]
[5]	[5]	[6]	[0]	[1]	[2]	[3]	[4]
[6]	[6]	[0]	[1]	[2]	[3]	[4]	[5]

Multiplication for table $n=7$

$⊝$	[0]	[1]	[2]	[3]	[4]	[5]	[6]
[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]	[4]	[5]	[6]
[2]	[0]	[2]	[4]	[6]	[1]	[6]	[5]
[3]	[0]	[3]	[6]	[2]	[5]	[1]	[4]
[4]	[0]	[4]	[1]	[5]	[2]	[6]	[3]
[5]	[0]	[5]	[3]	[1]	[6]	[4]	[2]
[6]	[0]	[6]	[5]	[4]	[3]	[2]	[1]

Addition table for  n=12

Consider the given value

for addition and multiplication, and drawing the table:

$\oplus$

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Multiplication for table $n=12$

$⊝$

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