# Can one usefully apply the Boolean functions of Mathematica to measurable Boolean sets?

This is something of an attempted succinct/pointed rephrasing of an earlier question Given measures on sets and on certain Boolean combinations of the sets, can one check their consistency and/or extend them to other combinations? (also posed on math.stackexchange Given measures on sets and on certain Boolean combinations of the sets, can one check their consistency and/or extend them to other combinations? )

I have various Boolean combinations of sets to which I am assigning—through exact and numerical integrations over them—(probability) measures $$\in [0,1]$$ (https://projecteuclid.org/euclid.pjm/1103038890 and https://math.stackexchange.com/questions/379203/measure-on-boolean-algebra), rather than simply True or False.

Is there a Mathematica framework in which one can check the consistency of such assignments and/or deduce, if possible through implication, measures on additional Boolean combinations? Is the Boolean suite of commands useful in this context—or would other methods be more appropriate?

## 1 Answer

There are essentially $$8$$ variables here:

$$\mu(A\wedge B\wedge C), \mu(\neg A\wedge B\wedge C),\mu(A\wedge \neg B\wedge C), \mu(A,\wedge B\wedge \neg C),\mu(\neg A\wedge\neg B\wedge C), \mu(\neg A\wedge B\wedge \neg C),\mu(A,\wedge \neg B\wedge\neg C),\mu(\neg A\wedge \neg B\wedge \neg C),$$ where $$\mu$$ denotes the measure. Lets label them $$A_1,\dots,A_8$$. They are subject to the constraints $$A_i\geq 0, i=1,\dots 8,$$ $$\sum_{i=1}^8A_i=1.$$

Now suppose you already know some boolean combinations. E.g. from your mathoverflow post, you know for example $$A\wedge B$$ (lets call $$P=A, S=B, PPT=C$$). But $$\mu(A\wedge B)=\mu(A\wedge B\wedge C)+\mu(A\wedge B\wedge\neg C).$$ So if you know $$\mu(A\wedge B)=x$$ for some constant $$x$$, then you get an additional constraint $$A_1+A_4=x,$$ where we recall that $$A_1= \mu(A\wedge B\wedge C)$$, $$A_4=\mu(A,\wedge B\wedge \neg C)$$. In this way you get an additional linear constraint on the variables $$A_1,\dots,A_8$$ for every entry in your table on Mathoverflow. You can simply let Mathematica compute the set of solutions to the system of constraint you obtain. Once you have this set of solutions, you can compute any other boolean combination you are interested in. For exmaple, you mention that you are interested in $$\mu(PPT\wedge (P\vee S))=\mu(C\wedge (A\vee B))$$. But this is simply $$\mu(C\wedge (A\vee B)) = \mu(A\wedge B\wedge C)+\mu(A\wedge \neg B\wedge C)+\mu(\neg A\wedge B\wedge C)=A_1+A_2+A_5.$$ So once you found solutions to the system of constraints, you can use these solutions to find possible values of any other remaining boolean combination by expanding it into the variables $$A_1,\dots,A_8$$.

• Thanks, I will apply this line-of-reasoning to the indicated problem. Rather bizarre situation/timing with respect to the mathoverflow posting mathoverflow.net/questions/359986/… It was indicated there that there would be "private feedback" regarding the closing of the question--but, at this point, I have not received any. May 13, 2020 at 15:33
• In the listing of the eight variables at the very beginning of the answer, the fourth and seventh have commas appearing in their arguments. I take it that these are simply typos. Also, there's a comma again in A_4 later in the text. May 13, 2020 at 15:40
• Concerning the concluding remarks: (1) what is the general procedure for expanding an arbitrary boolean combination into the eight "atoms", $A_1,\ldots,A_8$; and (2) how might one implement it in Mathematica? (I may post a mathematica.stackexchange question asking this.) May 14, 2020 at 10:35
• Per the previous comment, I did post this mathematica.stackexchange.com/questions/221893/… . Perhaps user250938 has some thoughts pertaining to that question. May 16, 2020 at 0:02