Thread: [SOLVED] addition and multiplication tables

I have these two Cayley tables. The addition table is complete and the multiplication table is not. The instructions say to use the distributive laws, but I can't seem to find the solutions using the distributive laws. I can only find the solution using the definition of a group.

Can someone help me?

ring R = {a, b, c}

+ a b c
a a b c
b b c a
c c a b

x a b c
a a a a
b a c
c a

Thanks a bunch!

The additiion table shows that is a group with identity element This is the zero element of

To complete the table for multiplication, take for example From the addition table, Hence, (You know from the partially completed table.)

Do the same for and (Hint: It will turn out that is the multiplicative identity, and the ring is a field with 3 elements.)