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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 to write boolean expression as linear equations 2

I just posted How to write boolean expressions as linear equations and asked about a simple example. Here's what we know so far: Suppose a,b,c,d,e ∈ {0,1}. if the boolean expression is: a ≠ b, I could use the linear equation a+b=1. if the boolean…
user66360
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1 answer

Dense Boolean SubAlgebras, equivalent definition

Definition (Dense Boolean SubAlgebra): Let $A$ be a Boolean algebra and $B\subseteq A$ a Boolean subalgebra. We say that $B$ is dense in $A$ if for every $a\in A$ there exist $b\in B$ such that $b \leq a$. I have found here…
albatros
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Simplifying $A'B'C'D+A'B'CD+A'BC'D'+A'BC'D+A'BCD'+AB'C'D'+AB'C'D+ABC'D'+ABCD'+ABCD$ to $A'B'D+A'C'D+AB'C'+ABC+BD'$ (and another)

I have been at this for hours using boolean algebra (not K-maps) and always end up so close but cannot reach the solutions. This is one expression: $$A'B'C'D + A'B'CD + A'BC'D' + A'BC'D + A'BCD' + AB'C'D' \\+ AB'C'D + ABC'D' + ABCD' + ABCD…
user961627
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A non embeddable boolean algebra

I am starting to learn about boolean algebras and one excercise is to find two boolean algebras $A$ y $B$ such that $\# A\leq \# B$ but there is not an embedding from $A$ to $B$. One example of a boolean algebra which is not a subalgebra is…
xyz
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Derive $p\lor (p\land q)\equiv p$ from the usual boolean algebra axioms

I'm reading Susanna Epp's book on discrete mathematics. In the section on logical equivalences she gives 11 laws, of which the first five are: $$Commutative: p\land q\equiv q\land p $$$$Associative: (p\land q)\land r\equiv p\land (q\land r)$$…
Erik Eriksson
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boolean algebra simplification, maybe a typo in the book?

I have found the following exercise: I have tried to solve it, but in any way that I try I cannot get the answer. I did the following: (C+A)(¬A+B)(C+B) Conmutative in terms (1) and (3) (C+AB)(¬A+B) Distributive So it says that…
Lila
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Expressing function for only using NAND and AND+NOT, is this correct?

First of all, sorry for my bad english. I'm not very comfortable writting about Boolean algebra in english but I'll try my best. So my teacher came up with this problem. Express the following function using only AND + NOT gates, then, NAND only. The…
Albert Lorenzo
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What is the correct way to draw a Karnaugh map

I'm studying digital logic at university currently, and the textbook has an example of a 4 variable K-map drawn with the $x_1$ and $x_2$ on top. However, most places I can find draw the map inverted with the $x_1$ and $x_2$ on bottom. I've taken to…
richbai90
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Let A be a Boolean algebra that has infinitely many atoms Let X be the set of atoms. Define the map from A to ℘(X) as follows, a → {x ∈ X : x ≤ a}.

Let $A$ be a Boolean algebra that has infinitely many atoms. Let $X$ be the set of atoms. Define the map from $A$ to $\wp(X)$ as follows, $a \mapsto \{x \in X : x \leq a\}$. (a) $f$ is 1-1. (b) $f$ is an isomorphism (c) $f$ is surjective (d) None of…
D0mBas3
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Boolean functions satisfying a condition

What is the total number of Boolean functions $f :\{0,1\}^n \rightarrow \{1,-1\}$ such that \begin{equation}\tag{1}\label{e} \sum_xf(x)f(x+y) = 0 \end{equation} for all $0\neq y$? Is there some closed form construction to obtain them all? Or even…
user 1987
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simplify: $AB(C+BD')+(AB)'$ boolean algebra

My solution $S= AB(C+BD')+(AB)'$ $S= ABC + ABD' + A' + B'$ How do I proceed from this ?
berserker
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How to solve this boolean algebra?

I tried to solve this $(x + y) (xy'z + xyz + xy'z')$ I got $x(xz + xy'z' + yz)$ as my final result How can I solve this?
Apurba Adhikari
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How to simplify from $\bar{c}(\bar{a}+b)+a$ to $\bar{c}+a$

I'm currently simplifying the current equation $$a.b+a.b.\bar{c}+a.\bar{b}+\bar{a}.b.\bar{c}+\bar{a}.\bar{b}.\bar{c}.$$ These are the steps I've taken so far: $$\bar{c}(a.b+\bar{a}.b+\bar{a}.\bar{b})+a(b+\bar{b})$$ and $b+\bar{b}$ simplifies to one.…
Ryan_DS
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Is this a complete and/or atomic subalgebra of $2^{2^S}$?

For some infinite set $S$, let $W:=\mathcal{P}(S)$ $B:=\mathcal{P}(W)$ $F:= \{p\in B: \exists s\in S\text{ s.t. }p=\{w\in W:s\in w\}\text{ or }p=\{w\in W:s\not\in w\}\}$ $A:= \{p \in B: \forall X\subseteq F(\bigcap X\neq\emptyset\text{ and }\bigcap…
Jeremy
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Boolean Algebra - Simplifying a Function

Simplify the function $f(x_2, x_1, x_0) = f(x) = \overline{\overline{x_0} \cdot x_1 + x_2 + x_1}$ such that the final minimized version only contains literals and the operators $"+" \text{and} "\cdot"$. Here's my answer: $f(x) =…
Monika
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