How to define equivalency as NAND only function?

Question

I am struggling a bit with boolean algebra. I need to represent equivalency as NAND only function.

$(A * B) + (-A * -B)$

I am trying with the Morgan rule but I don't know if I can do that:

$(A * B) + (-A * -B) = --(((A * B) + (-A * -B))$

Answer 1

I will use "$N$" for the NAND function.$$ab+a'b' = ((a'+b')(a+b))'=(a'+b')N(a+b)$$ Now, $$a'+b'=(ab)'=aNb$$ and $$a+b=(a'b')'=(a')N(b')$$ and finally $$x'=(xx)'=xNx$$ so we get $$ab+a'b'=\Big(aNb\Big)N\Big((aNa)N(bNb)\Big)$$

Answer 2

You can write any logical function with nands, because Not(x)=Nand(x,x), And(x,y)=Not(Nand(x,y))=Nand(Nand(x,y),Nand(x,y)) and finally Or(x,y)=Not(And(Not(x),Not(y)))=Nand(Nand(x,x),Nand(y,y)).

So: x=y=Or(And(x,y),And(Not(x),Not(y)))=Nand(Nand(Nand(Nand(x,y),Nand(x,y)),Nand(Nand(x,y),Nand(x,y))),Nand(Nand(Nand(Nand(x,x),Nand(y,y)),Nand(Nand(x,x),Nand(y,y))),Nand(Nand(Nand(x,x),Nand(y,y)),Nand(Nand(x,x),Nand(y,y)))))

Answer 3

Simply define all basic operations using NAND. Then given any complex boolean function you can subsitute their definitions (and bore yourself to death).