Question

Construct the operation table for Z₃ X Z₃ then write the inverse of each element. The following defines how to add two elements (x₁, y₁), (x₂.y₂) in Z₃ X Z₃:

(x₁,y₁)+(x₂.y₂)=(1+2 mod 3, y₁ + y₂ mod 3)

Answer 1

The operation table for Z₃ X Z₃ can be constructed by adding the elements using the given operation. The inverses of each element can also be determined.

Step-by-step explanation:

The operation table for Z₃ X Z₃ can be constructed by adding each pair of elements using the given operation. Here is the table:

(0, 0)
(0, 1)
(0, 2)


(0, 0)
(0, 1)
(0, 2)


(1, 0)
(1, 1)
(1, 2)


(2, 0)
(2, 1)
(2, 2)

To find the inverse of each element, we need to find a pair (x', y') such that (x, y) + (x', y') = (0, 0) for each element (x, y). Here are the inverses:

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• The inverse of (0, 0) is (0, 0)
• 
  
• The inverse of (0, 1) is (0, 2)
• 
  
• The inverse of (0, 2) is (0, 1)
• 
  
• The inverse of (1, 0) is (2, 0)
• 
  
• The inverse of (1, 1) is (1, 2)
• 
  
• The inverse of (1, 2) is (1, 1)
• 
  
• The inverse of (2, 0) is (1, 0)
• 
  
• The inverse of (2, 1) is (2, 2)
• 
  
• The inverse of (2, 2) is (2, 1)