Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [boolean-algebra]

Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Use this tag for questions about Boolean algebras as structures, or about functions defined from/to Boolean algebras. For Boolean logic use the tag propositional-calculus.

Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Use this tag for questions about Boolean algebras as structures, or about functions defined from/to Boolean algebras.

A Boolean algebra uses Boolean variables, typically denoted by capital letters, e.g. $A,B$, which can only take the values $0$ or $1$. Operators are $\land$ (conjunction), $\lor$ (disjunction) and $\lnot$ (negation).

For Boolean logic use the tag .

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Is this really a categorical approach to integration?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical Approach to Integration". The Abstract: "We…
Shaun
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Still struggling to understand vacuous truths

I know, I know, there are tons of questions on this -- I've read them all, it feels like. I don't understand why $(F \implies F) \equiv T$ and $(F \implies T) \equiv T$. One of the best examples I saw was showing how if you start out with a false…
user525966
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how many semantically different boolean functions are there for n boolean variables?

In short, this is an assignment question for a course I am taking - the exact wording is this: "Given n Boolean variables, how many 'semantically' different Boolean functions can you construct?" Now, I had a crack at this myself - and got pretty…
Zack Newsham
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Universal binary operation and finite fields (ring)

Take Boolean Algebra for instance, the underlying finite field/ring $0, 1, \{AND, OR\}$ is equivalent to $ 0, 1, \{NAND\} $ or $ 0, 1, \{ NOR \}$ where NAND and NOR are considered as universal gates. Does this property, that AND ('multiplication')…
Dilawar
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Any two points in a Stone space can be disconnected by clopen sets

Let $B$ be a Stone space (compact, Hausdorff, and totally disconnected). Then I am basically certain (because of Stone's representation theorem) that if $a, b \in B$ are two distinct points in $B$, then $B$ can be written as a disjoint union $U…
Qiaochu Yuan
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Prove XOR is commutative and associative?

Through the use of Boolean algebra, show that the XOR operator ⊕ is both commutative and associative. I know I can show using a truth table. But using boolean algebra? How do I show? I totally have no clue. Any help please?
Lawrence Wong
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Is XOR a combination of AND and NOT operators?

I'm not sure whether this is the best place to ask this, but is the XOR binary operator a combination of AND+NOT operators?
Dark Star1
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Duality principle in boolean algebra

All the definitions I came across so far stated, that if a statement is true, then also its dual statement is true and this dual statement is obtained by changing + for ., 0 for 1 and vice versa. However when I say 1+1, whose dual statement…
jcxz
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De-Morgan's theorem for 3 variables?

The most relative that I found on Google for de morgan's 3 variable was: (ABC)' = A' + B' + C'. I didn't find the answer for my question, therefore I'll ask here: What is De-Morgan's theorem for (A + B + C)'?
Billie
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Why are Boolean Algebras called "Algebras"?

Boolean algebras aren't algebras (to the best of my understanding). So why are they called algebras? Wouldn't it make more sense to call them a "Boolean system" or a "Boology" or something else like that?
Chill2Macht
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Find DNF and CNF of an expression

I want to find the DNF and CNF of the following expression $$ x \oplus y \oplus z $$ I tried by using $$x \oplus y = (\neg x\wedge y) \vee (x\wedge \neg y)$$ but it got all messy. I also plotted it in Wolfram Alpha, and of course it showed them,…
randomname
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What are some theorems made easier by Stone Duality?

I have seen a lot of praise for the Stone Duality Theorem, which links the algebraic structure of boolean algebras to the topological structure of stone spaces by a (contravariant) adjoint equivalence of categories. What are some theorems which are…
user788935
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What are the algebras of the double powerset monad?

Let $\mathscr{P} : \textbf{Set} \to \textbf{Set}^\textrm{op}$ be the (contravariant) powerset functor, taking a set $X$ to its powerset $\mathscr{P}(X)$ and a map $f : X \to Y$ to the inverse image map $f^* : \mathscr{P}(Y) \to \mathscr{P}(X)$. By…
Zhen Lin
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Examples of topologies in which all open sets are regular?

An open subset U of a space X is regular if it equals the interior of its closure, as we learn from the Wikipedia glossary of topology. Furthermore, the regular open subsets of a space (any space) form a complete Boolean algebra. I'm coming to this…
MikeC
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Stone's Representation Theorem and The Compactness Theorem

If you're working on $\mathsf {ZF}$ and you assume the compactness theorem for propositional logic, then you have the prime ideal theorem, and thus you can show that the dual of the category of Boolean algebras is equivalent to the category of…
Camilo Arosemena-Serrato
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