Are $(A \land \lnot B)$ and $(\lnot A \land B)$ universal gates?

Question

Are $(A \land \lnot B)$ and $(\lnot A \land B)$ universal gates?

(That is if we look at (A AND NOT B) as one individual gate, and (NOT A AND B) as another individual gate). Obviously these consists of two gates each, but im wondering if these are universal if they were integrated as a logic function. They both are one of the possible 16 "gates" that are available from a two input one output logic function, since $2^4 = 16$ possibilities.

Known universal gates are the NAND and NOR gate. NOR is number $1$ and NAND is number $7$ of the possible gates. They takes two inputs each, which the above boolean expression does also. I know to prove they are universal we can buildt other gates from them, I havent tried yet. But I also would like to know how to exactly prove wether the gates are universal, what are the most common methods to do this?

Answer 1

No, $A \land \lnot B$ is not universal (or functionally complete to use the standard terminology in mathematical logic). In the Wikipedia article in that link, you will find information about a general characterization due to Post of the functionally complete sets of connectives. (The statement of this characterization is a little intricate and the proof is quite hard.)

$A \land \lnot B$ fails because it preserves falsity: if $A$ and $B$ are both false then so is $A \land \lnot B$. As $\mathsf{false} \Rightarrow \mathsf{false}$ is true, there is no way of defining implication using just $A \land \lnot B$.

The same remarks apply to $\lnot A \land B$.