So I am given two binary operations $\times$ and ${+}$, such that given two values, say $A$ and $B$, $A{\times}B$ would equal $C$, where $C$ is the maximum value between $A$ and $B$, and $A{+}B$ would map to $D$ where $D$ is the minimum value for $A$ and $B$.
I know in order to prove that it is a boolean algebra I need to show that it follows the following five properties: Association, Distribution, Commutation, and that there exists an Identity and a Complement.
It has been given to me that the operations are NOT boolean algebras and it falls on me to prove why, but I'm not sure if I am approaching it incorrectly or what but I cannot seem to prove it. Any hints/suggestions?
Hint:
Let $S = \{0,x,1\}$, where $0 < x < 1$.
Check the axioms.
One of them fails.