I'm unable to convert this equation to NAND only.

Question

I've been trying to convert this equation $$ S_1\cdot S_3+\bar{A} $$ to NAND only but everytime i do it i get a different result, i will be glad if someone showed me the right way to properly solve this!

Thanks!

Answer 1

Hint: let me write $A \downarrow B$ for the NAND of $A$ and $B$ and use the more modern notation $A \land B$, $A \lor B$ and $\lnot A$ for conjunction, disjunction and negation respectively. Then for any $A$ and $B$ you have $$ \begin{align*} \lnot A &\equiv A \downarrow A\\ A \land B &\equiv \lnot(A \downarrow B)\\ A \lor B &\equiv \lnot(\lnot A \land \lnot B) \end{align*} $$ Using the last rule, get rid of all the disjunctions, using the second rule, get rid of all the conjunctions and, finally, using the first rule, get rid of all the negations. (You can optimise a bit using rules like $\lnot\lnot A \equiv A$, as you go.)