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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 .

3083 questions
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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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All finite boolean algebras have an even number of elements?

This seems obvious but I wanted to check, since I don't see it mentioned anywhere. If we define a boolean algebra as having at least two elements, then that algebra has a minimal element (0) and a maximal element (1). Since each element has a unique…
MikeC
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Example of Boolean Algebra that satisfies distributive law but violates complete distributive law

More precisely, I'm interested to know the example of Boolean Algebra $B$, such that for any $a, b, c \in B$, $a \cap (b \cup c) = (a \cap b) \cup (a \cap c)$, but there exists $\{ P_{ij}:i\in I, j \in J\} \subseteq B$, $\bigwedge_{i \in I}\vee_{j…
Metta World Peace
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2 answers

How to deal with an 8 variable Karnaugh map

I'm reaching back into my high school days trying to remember one of the rules about Karnaugh Maps. I have an 8 variable input, and I remember that I should try and make the selections a big as possible. However, I vaguely remember that if I have…
Adrian
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Existence of surjective homomorphism between Boolean algebras $\Lambda\subset\mathscr P(\mathscr B)\to\mathscr B$ (in ZF)

I am trying to prove the following theorem, due to Tarski according to W. A. J. Luxemburg on Reduced powers of the real number system and equivalents of the Hahn-Banach extension theorem: Given a Boolean algebra $\mathscr B$, there is a subalgebra…
user55268
8
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4 answers

Non-isomorphic atomless Boolean algebras

All countable atomless algebras are isomorphic. Can one give an example of a pair of mutually non-isomorphic atomless Boolean algebras of cardinaliy continuum?
MarkNeuer
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Non-isomorphic countable Boolean algebras

I'm trying to solve the next exercise: Construct a sequence $\mathcal{B}_0,\mathcal{B}_1, \ldots$ of countable Boolean algebras such that for all $m \neq n$ then $\mathcal{B}_m \ncong \mathcal{B}_n$. I know that two countable atomless Boolean…
natural
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Question on essential prime implicants

I am having some trouble understand essential prime implicants. So if a minterm is not covered by another overlapping rectangle, then that is an EPI. However, if we make a K-map for $f(x,y,z)=xy+xz'+y'z$, we have minterms m4, m6, and m7 not covered…
Snowman
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Definitionally equivalence between Boolean algebras and Boolean rings

On page 17, Introduction to Boolean Algebras,Steven Givant,Paul Halmos(2000): Motivated by this set-theoretic example, we can introduce into every Boolean algebra $A$ operations of addition and multiplication very much like symmetric difference…
Metta World Peace
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calculating number of boolean functions

I would just like to clarify if I am on the right track or not. I have these questions: Consider the Boolean functions $f(x,y,z)$ in three variables such that the table of values of $f$ contains exactly four $1$’s. Calculate the total number of…
Z Oj
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Determining don't-care values in a Karnaugh Map

I'm having a hard time understanding how to find the don't-care values in a Karnaugh map. What does it even mean? If I have a boolean function, say $f(a,b,c,d)=a'bc+abc'+bc'd+a'bc'd$, how would I determine don't-care values? What would I be looking…
Snowman
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Is there a connection between Boolean algebra and probability?

Is there a unifying abstraction that links Boolean algebra and probability theory? Both Boolean algebra and probability provide us the means to answer questions about set participation. On the one hand, Boolean algebra is an absolute, binary view of…
Dan Kowalczyk
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Boolean algebra question.

Is there a way to show that $$A\bar{B}C\bar{D}+D=A\bar{B}C+D$$ using the rules of boolean algebra? I tried several methods such as expanding D with $$D(D+\bar{D})$$ or adding $$D\bar{D}$$ to the equation but nothing worked. From the Karnaugh map it…
Vector_13
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Can someone explain consensus theorem for boolean algebra

In boolean algebra, below is the consensus theorem $$X⋅Y + X'⋅Z + Y⋅Z = X⋅Y + X'⋅Z$$ $$(X+Y)⋅(X'+Z)⋅(Y+Z) = (X+Y)⋅(X'+Z)$$ I don't really understand it? Can I simplify it to $$X'⋅Z + Y⋅Z = X' \cdot Z$$ I don't suppose so. Anyways, why can $Y \cdot…
Jiew Meng
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self dual boolean function

How many self-dual Boolean functions of n variables are there?Please help me how to calculate such like problems. A Boolean function $f_1^D$ is said to be the dual of another Boolean function $f_1$ if $f_1^D$ is obtained from $f_1$ by interchanging…
amitabha
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