       Re: Boolean algebra

• To: mathgroup at smc.vnet.net
• Subject: [mg69385] Re: [mg69373] Boolean algebra
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 10 Sep 2006 07:19:51 -0400 (EDT)

```On 10 Sep 2006, at 14:14, Andrzej Kozlowski wrote:

>
> On 9 Sep 2006, at 16:26, 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?
>>
>
>
> I seems to me that in general it will be pretty hard to use
> Mathematica for this sort of Boolean algebra without some AddOn
> package. However, your particular cases are very easy. Note that
> your assumptions amount simply to one assumption:
>
> In:=
> a2 = Not[a1];
>
> Now we have:
>
> In:=
> Simplify[ (b1 || a2) && (b1 || a1)]
>
> Out=
> b1
>
>
> In:=
> Simplify[(b1||a1||c1)&&(b2||a2||a1)]
>
> Out=
> a1||b1||c1
>
>
>
> Andrzej Kozlowski
> Tokyo, Japan

If you do not wish to use a global identity, you can use instead
Simplify with assumptions:

Simplify[(b1 || a2) && (b1 || a1), a2 = !a1 ]
b1
Simplify[(b1 || a1 || c1) && (b2 || a2 || a1), a2 = !a1]
a1 || b1 || c1

Andrzej Kozlowski
Tokyo, Japan

```

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