Is There an Order of Operations for Boolean Expressions in Sets?

[kmwest]

Hello all. Currently working on simplifying some Boolean expressions, one of the questions is:

( A int B U C) int B

I do not know how to go about simplifying the first term because there are not any parentheses within it and I have both the intersection and union symbols. Is there an order of operations, so to speak, where the union or intersect takes priority? Or I work left-to-right within a set of parentheses, etc. I've searched my textbooks and the Internet and I haven't found anything. Thanks in advance.

 

[micromass]

In my experience, an expression like A\cap B\cup C is ill-defined. It needs brackets to let it mean whatever you want it to mean.

So perhaps you copied this problem wrong?? Or the book makes an assumption that is nonstandard (in which case you should read the book to find it).

 

[kmwest]

There is a bar over the ( A int B U C) , another bar over the latter B, and another bar over everything. But I don't think that would affect the result.

I've done Boolean expressions before in a discrete math class (this is just an introductory chapter for a reliability engineering course) so this looks so unusual to me. I just re-read the section in the book and it has nothing like this, and only has the odd-looking expressions in the exercises.

 

[SteveL27]

There is a bar over the ( A int B U C) , another bar over the latter B, and another bar over everything. But I don't think that would affect the result.

I've done Boolean expressions before in a discrete math class (this is just an introductory chapter for a reliability engineering course) so this looks so unusual to me. I just re-read the section in the book and it has nothing like this, and only has the odd-looking expressions in the exercises.

I've never seen anyone write (A int B U C) because it's so ambiguous. But if I HAD to evaluate it, I would give intersection higher precedence than union, because intersection is analogous to multiplication, and union is analogous to addition. But I've never seen that convention used the way you have it here. I wonder if there's a typo.

Exactly what are the bars over? And are they bars as in "logical negation" or bars as in "topological closure?" Are you in set theory class or real analysis class? The more context you can give, the better.

 

[kmwest]

I just made the attached in Paint. This isn't even for a math class, it is part of a reliability engineering course. It is part of an early chapter on basic related math, including probability distributions and very introductory set theory.

 

[SteveL27]

I just made the attached in Paint. This isn't even for a math class, it is part of a reliability engineering course. It is part of an early chapter on basic related math, including probability distributions and very introductory set theory.

If it's not a typo in the book, take the intersection first. For the reason I have above. When people generalize intersections and unions to Boolean algebra, intersection is multiplication and union is addition. If you have to interpret this as opposed to asking your teacher if its a typo, then the only choice is taking the intersection first.

 

[kmwest]

Yes, I'm going to ask the teacher because nothing along these lines were in the text. I doubt it's a typo though, as it happens again in another question! Thanks.

 

[kmwest]

Interesting, when you type in 'b union c intersection a' into WolframAlpha, it produces a Venn diagram that is equivalent to 'b union (c intersection a)'. So you seem to be right, intersection should be taken first. (Prof hasn't got back to me yet...)

 

[AlephZero]

In my experience, an expression like A\cap B\cup C is ill-defined. It needs brackets to let it mean whatever you want it to mean.

So perhaps you copied this problem wrong?? Or the book makes an assumption that is nonstandard (in which case you should read the book to find it).

Within some fields (e.g. electronic logic circuit design) it is coventional to use the same notation conventions as the algebra of numbers, where addition is analogous to union and multiplication is analogous to intersection. So AB + CD means (A intersection B) union (C intersection D).

But I agree that if you don't know the context, the OP's notation is ambiguous.