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.

This is a Boolean ring, not algebra. – sqtrat Mar 23 '16 at 15:11 — sqtrat, Mar 23 '16 at 15:11

score 1 · Accepted Answer · answered Mar 23 '16 at 15:11

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).

answered Mar 23 '16 at 15:11

parsiad

25,154

Weird that it's different answer than what my Mathematica computes. BooleanMinimize[(C && (A || B)) || (D && (B || (A && C)))] // TraditionalForm yields $(A \wedge C) \vee (B\wedge C)\vee(B\wedge D)$. – Allstar Mar 23 '16 at 15:16

CAGT · Answer 2 · 2016-03-23T15:31:17.177

0

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 (+)

edited Mar 23 '16 at 15:31

answered Mar 23 '16 at 15:25

CAGT

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Simplifying a Boolean algebra equation

2 Answers2