Help me prove the following theorem about Mathematical Logic I am currently facing a problem where i need to prove that the NAND- and NOR-operators are the only operators that are a full basis by themselfs (so you can express every other operator by just using this one operator). I am half finished but stumbled upon the following lemma that I really need help on: 
Let $A \in \{1, 0\}$ and $\circ$ be a binary operator wich is a full basis of all other binary operators. Prove that there has to exist an Expression $\psi (A)$ that contains only $A$ and $\circ$, with $\psi(A) \equiv \neg A$
Note here that i want to ${\bf prove}$ that NAND and NOR are the only operators that are a full basis by themselfs. Therefore I can't use this information in my proof.
I hope I made clear what I need help on, so thanks in advance!
 A: The proof of the Lemma you refer to is as follows:
If $\circ$ provides a full basis for every binary operator, then it should be able to capture the $NAND$, and $A \ NAND \ A = \neg A$. 
Now, how is this a useful Lemma? 
Take the contrapositive: if some binary operator $\circ$ can't capture $\neg A$, then it does not provide a full basis for all binary operators (for example, it can't capture $NAND$). So: in your proof that $NAND$ and $NOR$ are the only operators that by themselves provide a full basis, you can rule out all the other ones that are not able to capture $\neg A$. Thus, for example $\land$ cannot provide a full basis, since any expression built up from only $\land$'s and $A$'s wil always evaluate to $True$ when $A = True$, and thus cannot capture $\neg A$. Indeed, all binary operators $\circ$ where $True \circ True = True$ or $False \circ False = False$ will not be able to provide a full basis by themselves. So that immediately rules out 12 out of the 16.
A: First, show that, for arbitrary formula $\phi$, it is impossible to show that $ \phi \equiv (\psi \circ \chi)$ where $\phi$ is some combination of variables of length 1 and $\neg$. 
Then show that any ($\phi \circ \psi$) can be expressed in some combination of $\neg$ and $\wedge$ or $\neg$ and $\vee$. 
Finally, show that NAND/NOR can be expressed in terms of $\neg$ and $\wedge$ or $\neg$ and $\vee$. 
It might work to show by induction on n that each n-ary bit function can be expressed using some combination of n distinct variables (maybe using some/all variables more than once) and $\neg$ and $\wedge$ / $\vee$.
