Change Boolean expression of XOR from AND-OR to OR-AND

Question

I have the Boolean of XOR as (ab') + (a'b), which is AND-OR; I need to turn it into an OR-AND.

I tried taking its complement, ((ab') + (a'b))' >> (ab')'(a'b)' >> (a'+ b)(a + b'), which does yield an OR-AND, but the truth tables dont align.

Answer 1

As Donald Splutterwit shows, the formal way to do this involves using the distribution laws. If you simply take the complement, then of course you can't expect the truth tables to align.

Here's an informal way to get the answer. As you note, $ab' + a'b$ is exclusive or: the expression is true when

[$a$ is true and $b$ is false] or [$a$ is false and $b$ is true].

That's clearly equivalent to "either $a$ or $b$ is true but not both $a$ and $b$ are true." Or, "either $a$ or $b$ is true, and either $a$ or $b$ is false. More formally,

[$a$ is true or $b$ is true] and [$a$ is false or $b$ is false].

More formally still, we can write $(a + b)(a' + b')$, which is the desired result.

Answer 2

Note that $a a'=0$ \begin{eqnarray*} ab'+a'b= aa'+ab'+ba'+bb'=a(a'+b')+b(a'+b')=(a+b)(a'+b') \end{eqnarray*}