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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 .

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What does it mean for a Boolean function to have a single minimal expression as SOP or POS.

My lecturer sometimes asks questions about functions that have a single minimal expression, I have trouble understanding what I can conclude from the fact that the expression is single. I know that a minimal expression consists of PI, but what does…
tom bros
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A sort of generalization of De Morgan's law for Boolean algebras

I've proved the De Morgan's law $\neg (x \vee y) = \neg x \wedge \neg y$. How can I prove that in every complete Boolean algebra $\neg \bigvee_{i \in I} x_i = \bigwedge_{i \in I} \neg x_i$?
effezeta
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Does always exist a homomorphism from a power set Boolean algebra to the 2-element Boolean algebra $\mathbf2$?

Wikipedia states: "there may exist many homomorphisms from a Boolean algebra B to 2". I would be very grateful for any references to the literature that there always exists a homomorphism from B to the 2-element Boolean algebra $\mathbf2$.
Victor M
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Expression for transitive matrices

I am trying to find a mathematical expression for the number of all possible transitive boolean matrices of order nxn. For example, $$T_1^n = n(n - 1)^3 + \frac{1}{6}n(n - 1)^4(n - 2) + \frac{1}{6}n(n - 1)(n - 2)(4n - 1)$$ This expresion T_1 works…
Jonathan
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Boolean XOR conversion- rewriting to XOR in simplifed form

I have a Boolean Expression simplified to: BC + AC + A'B'C' and get to this be rewriting: C(B + A) + A'B'C' I need to rewrite to (A + B) XOR C' I can check with a Truth Table and can show they are equivalent. I am having issues with the boolean…
user1560265
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In boolean algebra, can the property "ab + a'c = (a + c)(a' + b)" be considered consensus?

It has a form akin to the following: (a + t₁)(a' + t₂)(t₁ + t₂) = (a + t₁)(a' + t₂) at₁ + a't₂ +t₁t₂ = at₁ + a't₂ These are formally defined as "consensus", will it be accurate to describe the property mentioned in the title consensus as well?
Robert Cabrera
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How does the product of sum form work?

How does the product of sum form work? I understand that you take the product of the sums of the negated inputs of all rows of the truth table whose output is false. Taking XOR for example: A B Output 0 0 0 0 1 1 1 0 1 1 1 0 The…
FarmerZee
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Meaning of addition in the context of boolean algebra

In Boolean algebra, the sum refers to the XOR operation. This makes sense to me because in GF(2) the sum is supposed to implement a sum modulo 2, which has the same truth table of the XOR logical operation. However, in the context of sum of…
Giovanni Zaccaria
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What is the Schein rank of the Boolean matrix given below?

I am reading: Boolean Matrix Theory by K.H. Kim. On page 37 the definition for Schein rank is given. Def 1.4.1: For vectors $v,w$ the symbol $c(v,w)$ will denote the matrix $(v_i w_j)$. Such matrices are called cross-vectors. Def 1.4.2: The…
geoffrey
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Simplify $\bar{a}\bar{b}(\bar{a} + b)(\bar{b} + b)$

I've managed to simplify $\bar{a}\bar{b}(\bar{a} + b)(\bar{b} + b)$ to $\bar{a} +\bar{a}\bar{b}$ Where $\bar{a}$ = $a$ complement I simplified the original expression with Huntingtons Postulates, Idempotent laws etc I've checked they are equivalent…
Dravid
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Find the correct operator or function to satisfy a boolean equation

I have been working with a boolean model and got stuck on the following problem. I have a chain of boolean variables connected with an and like this: $a \land b \land c ... \land z$. I need to eliminate a particular variable from this chain using…
Mujtahid Akon
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Converting boolean expression to CNF

I need to convert an expression to CNF. I've converted it to DNF and I know its CNF, but i just can't do it by myself. I've tried the other answers, but with them it bloats to very long expression. It will took several hours just to write…
Ddystopia
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Is it possible to simplify the boolean expression $A'.B + A.B'$?

I know it's just an easy question but I can't get it at all. A'.B + A.B' Is it possible to simplify this expression? Actually, I got this expression from this A'.B.C+A.B'.C+A.B.C'+A.B.C = F C(A'.B+A.B') + A.B(C'+C) = F C(A'.B+A.B') + A.B =…
Python learner
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How to simplify this Boolean expression furthermore

Variables are $E, B, S, V_1, V_2, V_3$ Here is the Boolean expression that I need to simplify $$EB'SV_1'V_2'V_3 + EB'SV_1'V_2V_3 + EB'SV_1V_2'V_3 + EB'SV_1V_2V_3' + EB'SV_1V_2V_3$$ And this is how I simplified, $$\begin{align} &EB'SV_1'V_2'V_3 +…
Nureka Rodrigo
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boolean algebra: DeMorgan's law confusion

the following function should be put into table values: $$y = \overline{(a*b*d+c)}$$ So the first thing i am doing is using DeMorgan to get rid of the "whole-term-negation": $$y = (\tilde a + \tilde b +\tilde d * \tilde c)$$ from here all i do is…
ChrizZz
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