# Defining a Binary Operation

Algebra Level 5

A binary operation $$\otimes$$ on a set A, is a function which takes inputs $$a, b\in A$$ and produces an output $$a \otimes b \in A$$. For how many positive real values $$k$$ does there exist a binary operation $$\otimes$$ on the set $$[0,1]$$ such that the following properties hold for any $$x, y, z \in [0,1]$$:

$$\textbf{(A)}$$ $$x \otimes 1 = 1 \otimes x = x$$,

$$\textbf{(B)}$$ $$x \otimes ( y \otimes z) = (x \otimes y) \otimes z$$,

$$\textbf{(C)}$$ $$(zx) \otimes (zy) = z^k (x \otimes y)$$.

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