For context: I am learning Boolean Algebra by myself for fun and one of the questions in the book I am reading was a long boolean expression and the task was to simplify it to be the XOR boolean expression. I have managed to come quite a bit but cannot progress any further.
I am unable to simplify $\overline{(a+\overline{b})\cdot (\overline{a}+b)}$ to be $a\overline{b} + \overline{a}b$.
There seem to be no laws associated with this type of boolean multiplication. Can anybody point me in the right direction? My original equation is of an XOR logical operation, but not in its "final form" so to say.
Best Answer
XOR in conjuctive normal form is $(a+b)\cdot (\bar a+\bar b) = a\bar a + a\bar b + b\bar a + b\bar b$ by distributivity. But $a\bar a = 0 = b\bar b$ and $0+x = x$ for any Boolean expression $x$. Hence, the result is $a\bar b + \bar a b$ by commutativity.