# Universality of $a{\wedge\neg}b$ gate and similar

According to the Wikipedia and other sources only NAND/NOR make one-element sets that are functionally complete. Then, there are also a lot of two-element functional complete sets.
Question: why $a{\wedge\neg}b$ (and similarly ${\neg}a{\wedge}b$ and duals like $a{\vee\neg}b$ and ${\neg}a{\vee}b$) are not considered universal gates? Yes, they violate at least one of Post's criterion for universality (i.e. $a{\wedge\neg}b$ is falsity-preserving), but, is this so important? You can build a NOT gate and an AND gate out of one any above mentioned gates, thus, making them universal.
Where am I mistaken? :)

• You are mistaken. You cannot build a NOT gate, because NOT is not falsity-preserving ... Try building a NOT gate and you'll see. Notice that $a\wedge \neg a$ is equivalent to FALSE, not to $\neg a$. – Ewan Delanoy Aug 10 '17 at 8:45
• in a∧¬b set a to truth? the gate is, basically, a CNOT and a basis for universal computation in cellular automatons (i.e. glider collision is in accordance to this function) – artemonster Aug 10 '17 at 8:48
• You cannot set $a$ to TRUE unless you have previously shown TRUE to be in your generated system (and it isn't, since TRUE is not falsity-preserving ...) – Ewan Delanoy Aug 10 '17 at 8:50
• ooooooh. that means that my functionally complete set would be {truth, a∧¬b} ? – artemonster Aug 10 '17 at 8:50
• Yes, indeed. This one is functionally complete – Ewan Delanoy Aug 10 '17 at 8:51