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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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 ...
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910 views

Convert expression to NAND only

Endless youtube videos and reading through notes later I am yet again stuck. I have to covert the following to NAND only $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot ...
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76 views

Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
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66 views

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 ...
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Boolean Algebra Problem ABCC'

Hi I just want to ask the answer of this Boolean Algebra problem.. $$ABCC' + B + A'B $$ How to simplify that one?
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20 views

Boolean equation - bitwise AND operator

I have equation: (x AND B) XOR x = C where x - is unknown variable, B and C are constant. I need just one solution x that will satisfy this equation. How I can do this?
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algebra boleana and huntington axioms [closed]

Prove that there is an algebra onarit (that have only one member in the group "B") that meets with the Huntington axioms (without the axiom - at least 2 members in the group) edit: the aximos of ...
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1answer
29 views

Universal 2-bit gates

I'd like to show that there is no set of 2 bit reversible gates which is universal. I'm not sure as to where & how do I start here? I tried to assume by contradiction that such a set exists, thus ...
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3answers
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simplifying using Boolean Algebra.

I was doing the following question. Using the following rules of boolean algebra: _ law 1: X+X=1 law 2: X.1=X law 3:X.Y+X.Z = X.(Y+Z) simplify: ...
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1answer
31 views

Boolean functions

Boolean function f(x1,x2,x3): If f(x1,x2,x3)= TRUE then f(TRUE,x2,x3)= TRUE ...
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1answer
50 views

Counting switching functions

By using 16 bit binary in BCD , how many switching functions can exist ? Now , since this is BCD anything above 1001 is invalid. Considering 16 bits : 1001 1001 1001 1001 Above is number of possible ...
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1answer
76 views

Quotient of Cohen forcing

How do we know that the quotient of the Boolean algebra associated with Cohen forcing by a generic filter is either atomic or isomorphic to the Cohen forcing? I know that Cohen forcing is the unique ...
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2answers
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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 ...
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Question on M-generic filter

Let B a complete boolean algebra and $b, c\in B$ and M a model of ZFC. Why do we have that if $c\in G\,\, \forall G$ M-generic ultrafilter such that $b\in G$ then $b\le c$ ?
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Boolean algebras without atoms

Why is the theory of Boolean algebras without atoms $\omega$-categoric?
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1answer
14 views

Simple question on predense set in a boolean algebra

Let B a complete boolean algebra and D a subsets of B. Then D is predense below $ b\in B $, i.e. the downward closure of D is dense below b, iff $b\le \bigvee D$.Proving this equivalence seemed like ...
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1answer
50 views

Any two countable atomless Boolean algebras are isomorphic

How to prove that Any two countable atomless Boolean algebras are isomorphic. This is an exercise of Jech - Set theory book for which I have some difficulty.
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Boolean Algebra, 4-variable Expression Simplification

I have the following Boolean expression: $$w'x'y'z + wx'y'z + xz + xyz'\tag{1}$$ Upon doing my own work, I can only get as far as: $$zx + xy + zy'\tag{2}$$ Now, when I put the original equation ...
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FOUR-algebra - boolean algebra?

Belnap’s logic contains the the truth values 'true' ($t$), 'false' ($f$), 'unknown' ($\bot$) and 'paradox' ($\top$). Each of these is represented by a pair of bits: \begin{align} t &\rightarrow ...
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27 views

How does Absorption work in boolean algebra?

So I understand the basic outlines of the property: $a(a+b)=a$ From that its pretty clear to me that no matter what $b$ is the result will be $a$ regardless. However I don't understand how that ...
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684 views

Simplify the following expression using Boolean Algebra into sum-of-products (SOP) expressions

Simplify the following expression using Boolean Algebra into sum-of-products (SOP) expressions $Q.S.U + (Q' + S').(R + V) + U.(R + V) + Q' + S.T.U$ $.$ = AND $+$ = OR This is what I have so far ...
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1answer
63 views

Is Belnap's four valued-logic a boolean algebra?

Belnap's logic contains the the truth values 'true' (t), 'false' (f), 'unknown' ($\bot$) and 'paradox' (T). Each of these is represented by pair a of bits: t $\rightarrow$ (1,0) f $\rightarrow$ (0,1) ...
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31 views

boolean algebra - belnap logic

How to find out wether an algebra is a correct boolean algebra? So if we have the following algebra (rejects to belnap-logic theorems): $ \langle \{ w,f, \top , \bot \} , \wedge \vee \neg \rangle$
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Construct an OR gate when missing input information

Is there a way to construct an OR gate when the input for one combination is unknown? For example, suppose that the gate, $X$, outputs for the following inputs, $x_1$ and $x_2$, $x_1 = T$, $x_2 = ...
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How many truth tables if there are only $\land$ or $\lor$ for $n$ variables?

For example, if we have three operators $\land, \lor$ and $\neg$. For $n$ variables, there will be $2^{2^n}$ different truth tables. Because for $2^n$ rows of the truth table, there are $2$ choices - ...
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3answers
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How to convert a truth table to boolean expression?

If I have a huge truth table, it's hard for me to construct an expression. I know a problematic method, the Disjunctive Normal Form. But I found that I cannot reduce the huge expression. ...
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Complexity of computing a posiform of a quadratic pseudo-boolean function

I am reading the chapter 13, Pseudo-Boolean functions, of Boolean Functions: Theory, Algorithms, and Applications by Crama et. al. In section 13.2, the authors introduce the idea of Posiform. The ...
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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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problem simplifying boolean algebra expression using consensus theorem

Please simplify this logic expression for me with helping boolean algebra : A'C'D + A'BD + BCD + ABC + ACD' I know that must use consensus theorem . my solve : STEP 1 : Terms 1 & 3 ...
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Boolean algebra consensus theory

I want to simplify $wxy + x'z + y'z + wz = wxy + x'z + y'z$ but I can't seem to use the consensus theorem at the right place. I tried factoring cases for $x$ and $x'$ and $y$ and $y'$ but I don't ...
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48 views

Finite boolean algebra can be embedded into $\mathcal P(n)$.

I am trying to show that every finite boolean algebra can be embedded into $\mathcal P(n)$ for some large $n$. Any hints?
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2k views

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 ...
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How do you find the minterm list of a boolean expression containing XOR?

Let's say I have a boolean expression, such as F1 = x'y' ⊕ z . How do I go about finding the minterm list for that expression? The method I've tried is to take each term, such as x'y' and z, ...
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Reduce Boolean Expression

Note: A B = A and B A + B = A or B The expression: r = a̅ c̅ b + a̅ c b̅ + a c̅ b̅ + a c b Simplify?
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Calculation of Shannon entropy given the mutual information of Binary strings

Suppose $A$ and $B$ two different binary strings of length $l$. Suppose the Mutual Information (https://en.wikipedia.org/wiki/Mutual_information) of $A$ and $B$ is known to be $I$. Now suppose ...
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Simplifying Boolean Function with Karnaugh Maps

Given the boolean function f(x,y,z) = xyz + xyz' + xy'z + xy'z' + x'yz + x'y'z + x'y'z' (where x' = not x) In a three variable Karnaugh Map: ...
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Boolean algebra-Boolean ring. Stone Theorem?

I am interested in knowing which theorem is responsible for the following statement: Every Boolean algebra can become a Boolean ring by taking the ring addition to be $A\oplus B = (A \land \lnot B) ...
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1answer
32 views

Stuck at simplifying boolean expression

I'm getting stuck at the following boolean expression. z + (x'y) + (xy') + (xt') + (yt') In my solutions its simplified and the (yt') term is gone. How do they simplify this? I really cant see ...
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reducing Boolean expression to minimum literals

I'm finding it tough to simplify these types of expressions. Here's my problem: $(a+b+c')(a'b'+c)$ I have to reduce this to the minimum number of literals. So far I've only broken it down to: ...
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116 views

The negation of an implication statement

$$\neg(A \Rightarrow B)\lor \neg B$$ Does this this expression simplify to:? $$\neg A\Rightarrow\neg B\lor \neg B$$ Which further simplifies to: $$\neg A\Rightarrow\neg B$$
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Why is the dual of a filter an ideal?

Jech's set theory, (3rd edition) says that if $F$ is a filter on $S$ Let $I = \left\{ {S - X: X \in F}\right\}$ then $I$ is an ideal of $S$ (dual to $F$). However, let $X,Y \subset S$, $X \in I$ ...
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1answer
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Simplifying a Boolean function from a Kernaugh Map

Given the three variable Karnaugh Map: x\yz 00 01 11 10 \___________________ 0 | 0 1 1 0 1 | 1 0 0 1 I am supposed to write a ...
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1answer
28 views

Boolean Algebra fundementals

A disjunction A OR B truth table has A , B , and A OR B but mine has A ,B C, with A or B or C could some please explain this
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1answer
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Boolean algebras, Stone theorem and being isomorphic to a field of sets

I'm a little bit confused about duality between boolean algebras and topological spaces or sets. I know the following theorem (which is due to Stone, as far as I know): Every boolean algebra $B$ ...
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Simplification to DNF

To get right to the point. I have written a test which required me to Simplify to DNF. And the following equation gives me trouble. Here is the equation: ...
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21 views

Boolean Alegebra De morgans rule 2

hi i am told to perform a simplification using demorgans rule 2. Here is the question ' = Equals Not B . (C + B')' I got B' + (C' + B'') then B' + (C' + B) Now i dont know where ...
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Boolean Algebra expanding using absorption

Hi I have a question regarding the absorption law. I was told that I cannot expand ab = ab + abc by writing ab = ab(1+c). However, I believe you can expand xy = xyz' + xyz by doing xy = xy(z' + z) . ...
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Precedence of nested NOTs in boolean algebra

I have the following equation: $y = \overline{\overline{\overline{x_{1} + \overline{x_{2}}} .x_{2}.x_{1}} + \overline{x_{3}.\overline{x_{1}+x_{2}} + x_{2}}}$ I'm trying to solve it in four ways: ...
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Identifying a SSOP (standard sum of products) expression…

Say you're asked to identify a standard sum of products (SSOP) expression from 4 or 5 options... 3 of them are definitely not SSOP (variables are missing between the terms)... however two of the ...
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proving properties of (graph) dominance defined via a system of equations

Some notions on graphs can be defined via a system of equations with values in a lattice. For example, dominance $d(v_1, v_0)$ ($v_1$ dominates $v_0$) in a graph $g$ is defined by a system $\forall ...