# Are there impossible boolean constructions?

I was reading about logic and I remember, for example: That with the binary $\mathtt{NAND}$ connector can be used to assemble all the other binary connectors - I already know that there are primitive $n$-ary connectors that could be used to assemble all the other connectors. This single tool is used to assemble all the other connectors.

In euclidean geometry, there are two tools, compass and straitedge which are used to make some possible constructions. Thinking about both I asked myself if it would make any sense of talking about boolean constructions, if there are constructions that are not possible to be made with the given boolean tools.

Although I have no idea of what such construction could be - which in the case of euclidean geometry there are some easy imagineable examples, for example: The quadrature of the circle - I don't know if someone has already thought about such objects.

I recognize that the question may be an exploration into vague and poorly defined ideas, but I'll be a lot happy for some feedback on it.

• For example it is not possible to create a non-monotonic function with only monotonic operators (e.g. $\mathtt{AND}$ and $\mathtt{OR}$ are monotonic, $\mathtt{NOT}$, $\mathtt{XOR}$, $\mathtt{NAND}$ are not). – dtldarek Oct 29 '13 at 4:42
• @dtldarek Thanks for the information. I'll read about it. I loved the \mathtt{}! – Billy Rubina Oct 29 '13 at 4:46