Simplifying a Boolean algebra equation

Question

I have a boolean algebra equation that i'm not able to simplify fully.

\begin{align} &(c+ab)(d+b(a+c))\\ &(c+ab)(d+ba+bc)\\ &cd+ abc + bc^2+abd+a^2 b^2 + ab^2 c\\ &\text{using boolean laws $x^2=x$ and $x+x=x$}\\ &cd + bc + abd + ab + (abc + abc)\\ &cd + bc + abd + ab + abc \end{align} And now I get stuck. Mathematica simplifies this to $ac+bc+bd$, but I just don't see how.

Answer 1

You might have typed this into Mathematica incorrectly. Here's the solution:

\begin{align*} (c+ab)(d+b(a+c)) & =(c+ab)(d+ab+bc))\\ & =cd+abc+bc+abd+ab+abc\\ & =cd+abc+bc+abd+ab\\ & =cd+bc+ab(c+d+1)\\ & =cd+bc+ab \end{align*}

Here's Wolfram Alpha computing the same thing (DNF).

Answer 2

You have backwards the AND's and the OR's entered in Mathematica, you should use:

BooleanMinimize[(C || (A && B)) && (D || (B && (A || C)))

|| = AND (*)

&& = OR (+)