Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Questions regarding Boolean algebras as structures, or regarding functions defined from/to Boolean algebras fit into this tag very nicely. For Boolean logic use the tag propositional logic

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Boolean Simplification: (A+C)(!A+B)(B+C) = BC

How might I solve this? I can't find any problem similar to this, and I always end up with the wrong terms. If (AB) = 0 and (A+B) = 1, prove that (A+C)(!A+B)(B+C) = BC
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Boolean Algebra

There seems to be some discrepancy between my answer and the solution's. Can somebody please tell me what I have done wrong? Thanks! $$\begin{align} \left(A \lor B\right) \land \left(B \lor C\right) ...
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toggle boolean and more

Recently I used this to toggle a Boolean value, b being the current value and self.status being the result self.status = (b-1)*(b-1) This rather than use an if ...
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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 ...
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is the following a Boolean Algebra?

Boolean Algebra: $$D_{30}=\{n:n\mid30\}= \{1,2,3,5,6,10,15,30\}$$ I don't know how to test that this is a boolean algbra (a BA is a distributive lattice with $T,F$ in which every element has a ...
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Proving if equations are products of XOR

I have to prove analytically to see if these equations are exclusively or. $$A⊕ A=0$$ Do I solve this by using the truth table? ...
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Generating Input Binary Combination Dynamically

this is probably right forum to post this question I am currently working on a application where there is a requirement to generate binary combination of input signals in a truth table. The signal ...
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537 views

Show that $x \wedge y = x \iff x \vee y= y$ using the following properties

Suppose $L$ is a set with binary operations $\wedge$ and $\vee$, along with the special $T$ and $F$ (for an $x,y,z \in L$) such that the following rules hold: $x\wedge T=x$; $x\vee y=T$ ...
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How to prove distributivity in Boolean Rings

A Boolean ring is a ring which all of its elements are idempotent, i.e. $a ^{2}=a$. I know that If we interpret multiplication and addition in such a ring, as meet and joint respectively, then Boolean ...
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Boolean simplification

So I am giving this expression D +B’C’ + CD’ +A B’C and I ask to simplify it When working through it I get D+B'C'+CD'+AB'C D'(A'B'+CD'+AB) D'(A'B'+A(B'+B)) D'(A'B'+AC') D'(B'+A) Am I on the right ...
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Boolean Function

A Boolean expression is given: (A B)’ + B C’ +A’ C = F. Construct the logical circuit and draw the timing diagram of the output F. I am not sure where to start.
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Boolean Simplification

I'm having some trouble getting a handle with this course. We are starting Boolean algebra and my professor wants us simplify the following: (AB)'+(A'+B')'= (AB)'+BC+A'B'C'= I am assuming the "()" ...
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Adjoint for functor involving Boolean rings

Let $R$ be a a commutative ring with a unit element, then one can associate to $R$ a Boolean ring $B(R)$, as in this text by Bergman, last line of page 594. (I guess this is a very classical thing. ...
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Linearly separate a cube

I am facing the problem of the linear separability of a three dimensional cube. Let's take the opposite vertexes of the cube as $(0, 0, 0), (1, 1, 1)$. It is possible to split it with a plane in two ...
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Terminology question; inverse vs complement in Boolean algebra

This was said at a lecture I attended: $e$ is neutral element for operation $*$ if $\forall x (x*e=x \wedge e*x = x)$. So, for example 0 is n. e. for disjunction and 1 is n. e. for ...
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Help at solving boolean function.

I`m having some difficulties solving a boolean expression (I am converting it to CNF form). The expression is: $$F = (Q_1 \to P1 \land \lnot P_2) \lor Q_1 \land P_2 \lor P_1$$ So i do not know, ...
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Non-principal ideal in Boolean ring

Does anyone know a simple example of a Boolean ring with a non-principal ideal? Every finitely generated ideal in a Boolean ring is principal, hence such an ideal cannot be finitely generated...
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Boolean Logical Algebra - Prove 4 nor gates to an xnor gate.

Need to reach the following conclusion (or maybe its the premise?) AB + A'B' = F http://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/XNOR_using_NOR.svg/256px-XNOR_using_NOR.svg.png
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Infinite boolean sequence

I was given the following problem: Let $V_1, V_2, \dots$ be an infinite sequence of Boolean variables. For each natural number $n$, define a proposition $F_n$ according to the following rules: ...
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Boolean algebra spectrum

The first-order spectrum of a theory, is the set of cardinalities of its finite models. Finite models of Boolean algebras are informally n-dimensional cubes, therefore boolean algebra spectrum is the ...
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Boolean Algebra - Truth Table

X'Y'Z' + XYZ I have the equation above (Boolean Algebra truth table), and as I know, if I get x' and the value of x is 0, the value will change to 1. But Y' with the top bar being higher, what ...
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Number of canonical expressions

There is a question: What is the number of canonical expressions that can be developed over a 3-valued boolean algebra? I was trying to solve this. Canonical expression is the combination of ...
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666 views

Boolean Algebra Manipulation/Simplification

I have come across a couple questions while doing my digital logic work. 1) Is it possible to simplify these, while keeping each a product of sums? (I'm leaning towards no--the only way I could see ...
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80 views

Which of the following representations of Karnaugh map is 'better'?

I usually come across two representations of Karnaugh maps in books and on the web as shown in the figure. The difference is whether the higher order variables are on the rows or on the columns. I ...
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Boolean circuits and digraphs

It is well known that connecting NAND gates allows the construction of arbitrary circuits. Furthermore, a NAND gate can be represented as a digraph with four vertices (in order, the two inputs, the ...
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Boolean function simplification

I'm having problems with the following expression: (A'+B)'+B(A'+AC)+ABC' And here is what I tried to simplify: ...
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Better compression for a positive DNF than via BDD

I am experimenting with compressing positive disjunctive normal form (DNF). When I use binary decision diagrams (BDDs) related algorithms I don't get very good results. For example the algorithms ...
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Three questions on chapter 7 of Jech's Set Theory

In the proof of Pospisil's Theorem (theorem 7.6) that there are $2^{2^\kappa}$ uniform ultrafilters on $\kappa \geq \omega$, the author writes : Let $\mathcal{A}$ be an independent family of subsets ...
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136 views

Complete Lattice with unique negation

Suppose i have a complete lattice (meet and join exist for any two subsets of the lattice) and for every element of the lattice, there is a unique element that is its complement (hence a Boolean ...
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63 views

Number of basis representations of Boolean vectors

Suppose I have a basis $v_1, ..., v_n$ of boolean vectors and a boolean vector $v$ that was constructed by iterated $v = v_i$ $OR$ $v_j$ statements. I want a way to figure our how many unique ...
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Simplify $A'B'C'D' + A'B'CD' + A'BCD' + ABCD' + AB'CD'$

How do you simply the following equation? $$X = A'B'C'D' + A'B'CD' + A'BCD' + ABCD' + AB'CD'$$ Here is what I did: $$\begin{eqnarray} X & = & A'B'C'D'+A'CD'(B'+B) + ACD'(B+B') \\ & ...
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Localizations of the Boolean Ring P(X)

Given a set $X$, we can construct the Boolean ring whose elements are the power set of $X$. The multiplication therein is intersection, and the addition is symmetric difference. I am interested in ...
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Boolean Algebra simplification - odd number terms

I'm new to boolean algebra and having problems simplifying expressions with odd number terms, Expressions such as: 1. A'B'C'D + A'B'CD + AB'C'D + AB'CD + ABC'D 2. A'BC + AB'C' + A'B'C' + ...
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A question about the regular languages being closed under Boolean operation (how to generalize)

I know that if $L_{1},L_{2}$ are regular languages then so is $L_{1}\cap L_{2},L_{1}\cup L_{2}$ are regular languages, I also know that $L$ is regular $\implies L^{c}$ is regular . It is easy to ...
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Converting a Proposition to DNF using proof systems

I have been attempting to convent a prop to DNF using a group of common rules, i have applied them all but i think i should be able to get it smaller, This is what I've got so far. Thanks! $$(p \wedge ...
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how to solve system of linear equations of XOR operation?

how can i solve this set of equations ? to get values of $x,y,z,w$ ? $$\begin{aligned} 1=x \oplus y \oplus z \end{aligned}$$ $$\begin{aligned}1=x \oplus y \oplus w \end{aligned}$$ $$\begin{aligned}0=x ...
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Lindenbaum Algebras

After reading this page, I still have some questions about Lindenbaum algebras. Assume that the scope is a propositional language with a denumerable set X of propositonal variables. In that case, ...
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boolean algebra simplification to remove extra term

how do i simplify this equation using boolean algebra: AB + ¬AC + BC to be equal to AB + ¬AC the BC is unneeded, but how do i remove that term using boolean algebra?
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Saturated Boolean algebras in terms of model theory and in terms of partitions

Let $\kappa$ be an infinite cardinal. A Boolean algebra $\mathbb{B}$ is said to be $\kappa$-saturated if there is no partition (i.e., collection of elements of $\mathbb{B}$ whose pairwise meet is $0$ ...
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Do filters on a Boolean algebra also make a Boolean algebra?

Let $\mathfrak{B}=(B,\bot,\top,\lnot,\wedge,\vee)$ be a boolean algebra. $B_F$ be the set of all filters on $\mathfrak B$. And for all filter $F$, $G$, $F \wedge_{B_F} G \colon= \mathbf C(F \cup G)$ ...
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Lindenbaum algebra is a free algebra

The following is a continuation of this question. I would like to prove that the Lindenbaum algebra is a free algebra. Hopefully I would like to hear hints on how to proceed in the 'right' ...
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What do you call 2 boolean functions which are equivalent if two arguments exchanged?

What do you call boolean functions which are identical accurate to argument order? EDIT1 I meant not symmetric function. I mean, for example, implication function with truth table 00=1 01=1 10=0 ...
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271 views

Free boolean algebra

Consider the following definition: Let $X$ be a set and $e : X \mapsto A$ a mapping to a boolean algebra $A.$ We say that $A$ is free over $X$ (with respect to $e$) if for every mapping $f:X ...
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Cardinality of the set of ultrafilters on an infinite Boolean algebra

Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?
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Inducing maps between Boolean completions of posets

Given a separative poset $P$, we can form it's Boolean completion $B(P)$. This is a Boolean algebra whose elements are regular cuts on $P$, as defined here. Also, $P$ embeds densely into $B(P)$ by ...
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Stone duality for ideals and filters (exercise)

In A Course in Universal Algebra (Burris, Sankapannavar), the exercise 4.4.7-8, p.158, says: Let $A$ be a Boolean algebra. Denote $A^\ast:=\{\text{ultrafilters of }A\}$, and give $A^\ast$ the ...
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A free boolean algebra

Consider the following definition: The boolean algebra $A$ is generated freely with the subset $G \subseteq A$ if for every boolean algebra $B$ and map $f:G \mapsto B$ there is precisely one ...
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Automorphisms of a boolean algebra

Let $A = P(\mathbb{N})$ be the powerset of the natural numbers. We can look at $A$ as the Boolean aglebra - having in mind the obvious operations on elements of $A$. What I am interested in knowing ...
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Extracting information from simultaneous boolean AND, XOR, and NOT

I'm looking to extract some information from a series of equations with AND, XOR and NOT. I've already covered all of the easy parts using various boolean identities, so I'm looking to now determine ...
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Non-Boolean group with every element of order two

Let $G$ be a group (not necessarily finite) such every element of $G$ has order 2. Every such group is abelian [1]. Clearly, every Boolean algebra $B$ is a group of this type, when equipped with the ...