       Re: Boolean algebra

• To: mathgroup at smc.vnet.net
• Subject: [mg69426] Re: [mg69373] Boolean algebra
• From: "Carl K. Woll" <carlw at wolfram.com>
• Date: Tue, 12 Sep 2006 06:52:42 -0400 (EDT)
• References: <200609090726.DAA21734@smc.vnet.net>

```Menace wrote:
> I'd like to solve the following problem in Mathematica,
>
> Given are the following preliminaries:
> (a1 || a2)=True
> (a1 && a2) = False
>
> Then, given the following conjunctive normal form:
>
> (b1 || a1) && (b1 || a2)
>
> This can be simpliefied to: b1 || (a1 && a2)
>
> Given the prelininaries I'd like Mathematica to simplify this to: b1.
> However, I cant figure out how to do this. I tried the following:
>
> prelims := {(a1 && a2) -> False, (a1 || a2) -> True};
> f1 := (b1 || a2) && (b1 || a1);
> Simplify[f1] /. prelims
>
> and this seems to work. However
>
> f2 := (b1 || a1 || c1) && (b2 || a2 || a1);
> Simplify[f2] /. prelims
>
> evaluates to: (b1 || a1 || c1) && (b2 || a2 || a1) instead of b1 || a1
> || c1.
> The reason of this seems to be the order in which a2 and a1 occur in
> f2. How can I make sure it works regardless of the order of a1 and a2?

Here is one possibility that might be helpful:

cond = Not[a1&&a2]&&(a1||a2);
f1=(b1 || a2) && (b1 || a1);
f2=(b1 || a1 || c1) && (b2 || a2 || a1);

In:= Simplify[f1 && cond || ! cond, cond]

Out= b1

In:= Simplify[f2 && cond || ! cond, cond]

Out= a1 || b1 || c1 || ! a2

The first example could also be done with Simplify[f1,cond].
Unfortunately, in the second example the result isn't quite what you
wanted (as Andrzej said, variable elimination in Boolean algebras would
be needed to eliminate the !a2 term), and I suspect that this technique
will not be adequate for more complex examples.

Carl Woll
Wolfram Research

```

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