Which of the following formulae yield a welldefined a binary

Which of the following formulae yield a well-defined a binary operation * on Z_5? Here [a] indicates the congruence class of the integer a in Z_5. [a] * [b] = [|a + b|] for all a, b epsilon Z. [a] * [b] = [a^2b] for all a, b epsilon Z.

Solution

(a) This is not a well-defined binary operation as [1]*[2] and [-4]*[2] are different.

[1]*[2]=[|1+2|]=[3]...............(1)

whereas (note that [1]=[-4] ,as 1-(-4) =5 is divisible by 5).

[-4]*[2]=[|-4+2|]=[2].................(2)

As [2] and [3] are distinct elements in Z₅, it follows that the binary operation is NOT well-defined .

(b) This is a well-defined binary operation :if [a]=[c] and [b]=[d], then [a² b]= [c²d]

Proof: [a]=[c] means a =c+5k and [b]=[d] means b = d+5m for integers k and m.

Now a²b = (c+5k)² (d+5m) is clearly of the form c² d+5r, for some integer r.

So [a² b]= [c²d] in Z₅.

This establishes that the given binary operation is well defined on Z₅

, so [a]*[b]= [c]*[d],.

Thus * is a well-defined binary operation on Z₅.