Find all algebras with one binary operation and the set {0,1}

Question

I believe the only algebra is (A,*). +,-,/ don't meet the requirements. But how do I prove that there is no other? What are all the binary operations on numbers?

Answer 1

With no conditions, a binary operation is an arbitrary function $A\times A\to A$. There are $4$ elements in $A\times A$ and two elements in $A$, so there are $2^4=16$ possible binary operations. To enumerate them, write $$A\times A=\{(0,0),(0,1),(1,0),(1,1)\}$$ and assign all possible sequences of values to this.