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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Silly Question (monotonic) (updated)

suppose we have a measure $\mu$ on an algebra $B$, and $E,F\in B$ I know, if $E\subseteq F$, then $\mu(E)\leq\mu(F)$. Does the converse true, when $0<\mu(E)\leq\mu(F)$ ($\mu$ is nonatomic).
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Prove the following $f_{(A \cup B)}(x)=f_A(x)+f_B(x)-f_A(x)\cdot f_B(x)$

There is option to prove the following with truth table? $$f_{(A \cup B)}(x)=f_A(x)+f_B(x)-f_A(x)\cdot f_B(x)$$ I would like to get some hints how to do it in formal way(not truth table) thanks!
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1answer
36 views

Boolean Law that proves theorem

What Boolean Law proves the following theorem: (a && b) || (b && c) || (a && c) = (a || b) && (b || c) && (a || c) I made ...
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One implication (on Measure)

Please be noted that charges are finitely additive measures and measure are countably additive ones. Theorem 2.1. Let $\mu$ be a charge on a Boolean algebra $B$. Each of the following conditions ...
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DNF and CNF logic problem

So i want to find the DNF and CNF of : $ 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 and stuff, I also plotted it in ...
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141 views

Logic boolean algebra problem

so I have to prove that these equations : are equivalent?
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41 views

Must complete atomless Boolean algebras of the same cardinality be isomorphic?

More generally: must complete Boolean algebras of the same cardinality and with the same cardinality of atoms be isomorphic?
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85 views

Boolean algebra simplification problem

I can't solve this equation: $$(xy + x'yz)(xz + x'y') = xyz$$ After applying distribution I got this: $$xyz + yz + x'z = xyz$$ I can't find the answer and have been thinking for hours now.
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1answer
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Simplifying This Boolean Expression? (A Little Rusty)

I have the Boolean expression: F = A'B'C'D + A'BC'D' + ABC + AB'C'D' + ABCD'. Note that the ' indicates the negation of the variable by my convention. I am trying to show that F = BC + A'C' + B'D' is ...
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3answers
140 views

Can Boolean ring without unit be embedded into a boolean ring?

While going through a book (Lectures on Boolean algebra, Halmos) I got struck at the following question : Prove that every Boolean ring without a unit can be embedded in a Boolean ring with a unit. ...
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51 views

Boolean Algebra simplification, just cant get it

I have this question that I need some help with, I just can't get to grips with simplifying. I'm looking at the rules and such but I just can't see where to apply them. Can someone show me the ...
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0answers
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Expansion of subsets of a hamming ball in hypercube

Consider a hypercube graph $G_n = (V,E)$ in n dimensions. Let $H_{1/2} \subset V$ be the set which represents the hamming ball of radius $n/2$. That is for every $v \in H_{1/2}$ the hamming weight of ...
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2answers
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How to simplify $A(\overline BC+B)$

How do I go from $A(\overline BC+B)$ to $A(B+C)$? What definition should I use to get the final answer? Would like an explanation and proof so I can learn rather than just memorise.
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2answers
87 views

Solution to ax+b=c in a Boolean algebra

I have a question. In another forum, a user asked if there is a solution to ax+b=c in a Boolean algebra, where "ax+b=c" is "$(A \wedge X) \vee B = C$". The idea is that, in a Boolean ring, this ...
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23 views

is this boolean algebra transformation correct

I have the following expression: (A=1 or A=2) or (B=1 or B=2) and try to transfer it to: (A=1 or A=2 or B=1 or B=2) Are these two expressions equal?
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1answer
44 views

How can this boolean algebra equation be solved?

(7EFEFEFF + A) ^ (A ^ FFFFFFFF) = 81010100 How can it be solved for the value A? where '+' means plus '^' is bitwise XOR EDIT: Forgot to mention. A is a ...
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1answer
220 views

Solving special boolean equation set

I have boolean equation sets that look like this (where ^ means xor): eq 1: x1^x3^x5^x6^x9^x10^x11^x13^x17^x18 = 0 eq 2: 1^x1^x3^x10^x12^x17 = 0 eq 3: 1^x2^x3^x5^x8^x10^x14^x16 = 0 ...
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“Optimal Disjoint Decomposition” of a Boolean Lattice Subset?

I am looking for the name (and, possibly, an efficient solution) of the following problem: Given a Boolean lattice $(L, \sqcap, \sqcup)$ with least element $0$, and a finite subset $X \subseteq L$, ...
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2answers
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Simplify this Boolean Expression X + (~x) * y [closed]

Need help on this Homework. Don't know how to start
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75 views

2-Colorable & Decision Problem

Consider the following decision problem. Given $m$ subsets $A_{1}, \dots , A_{m} \subset \{1 , \dots , n \}$. Does there exist a subset $S \subset \{ 1, \dots ,n \}$ such that the cardinality of the ...
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1answer
58 views

Atomic Boolean Algebras that are not canonical

I define a Boolean Algebra B to be canonical iff it is isomorphic to the powerset of some set S. (under union, intersection, and complement, of course). Any Boolean Algebra that is both atomic and ...
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3-Colorability Graph Questions

I know that a boolean formula for 3-colorability is : $ \wedge_{i \in Vertices}(\bar{b_{i,1}} \vee \bar{b_{i,2}}) \wedge_{\left(i < j \right)\in Edges} ((b_{i,1} \bigoplus b_{j,1}) \vee (b_{i,2} ...
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2answers
100 views

Example of a function between boolean lattices that preserves $(\top,\bot,\wedge,\vee)$ but not complements.

Its easy to find boolean lattices $A$ and $B$ together with a function $f : A \rightarrow B$ such that $f$ preserves both top and bottom elements, as well as binary meets, but not complements. ...
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Simplifying this boolean function

How can I completely simplify this equation using algebraic simplification rules? $$x'y'z + x'yz + xyz$$
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2k views

Simplify Boolean Product of Sums Function

I've got a product of sums expression: F=(A'+B+C')&(A+D')(C+D') I need to show it as a sum of products and then simplify it. Right now I got: F=(A'&D')+(A&B&C)+(B&D')+(C&D') ...
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1answer
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Boolean function on $\{0,1\}^n$ comprising just binary AND and OR gates

Let $f:\{0,1\}^n\to\{0,1\}$ be a boolean function computed by logical circuit comprising just binary AND and binary OR gates (assume that the circuit doesn't have any feedback). Let ...
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1answer
87 views

Truth table to prove statements

A, B and C. When questioned A says ''If B did not do it, then it was C." B says ''A and C did it together or C did it alone". C says ''We all did it together." How would i be able to put these into ...
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1answer
41 views

Write $p \rightarrow \lnot q$ in CNF form with only and ,or and brackets

Write $p \rightarrow \lnot q$ in CNF form with only and, or, and/or brackets How on earth would I even do this? Completely lost! Any help appreciated.
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1answer
40 views

Boolean algebra (ISO)

Let $\mathfrak{A}$ be a Boolean algebra and $E$ be an element in $\mathfrak{A}$. The set of all subelements of $E$ forms a Boolean algebra, denoted by $\mathfrak{A}_E$. Suppose that $I$ be the ...
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98 views

Boolean formula vs boolean function.

Is there a technical difference between boolean formulas and boolean functions?
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How to simplify this using boolean algebra?

My paper is due tomorrow and there is only the last exercise left for me to do. However, I don't have any sufficient notes or examples on how to simplify it. Any help would be appreciated! A'B'C' + ...
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31 views

Explain one statement about Stone Space

In this page Stone Space This is no clear for me "The points in S(B) are the ultrafilters on B, or equivalently the homomorphisms from B to the two-element Boolean algebra" I think I means every ...
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1answer
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What is the underlying math in this relation?

Suppose we have the constraint $$.7x_1+.4x_2+.5x_3<1,$$ $$x_1,x_2,x_3\in\{0,1\}$$ Then we can convert it to a Boolean expression with binary variables of the form $$(\neg x_{1}\wedge\neg ...
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1answer
63 views

Can the obvious “product” of complete atomistic Boolean algebras be realized as a categorial product?

Let $X$ and $Y$ denote sets, and $\eta_X,\eta_Y : X,Y \rightarrow X+Y$ denote the natural injections to the disjoint union. Then intuitively, the "product" of the Boolean algebras $2^X$ and $2^Y$ ...
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Convert a Boolean expression to a linear expression?

Suppose we have a Boolean expression $$(\neg x_{1}\wedge\neg x_{2})\vee\left(\neg x_{1}\wedge\neg x_{3}\right),$$ which we need to be true. Is there a method to convert this to a linear expression of ...
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138 views

How can I get a product-of-sums from this sum-of-products?

I have this function: $$f(A,B,C,D) = A'B' + CD' + ABC + A'B'CD' + ABCD'$$ I used a Karnaugh map to minimize the function to: $$Minimum SOP = A'B' + C D' + A B C$$ How can I turn this into a ...
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Represent boolean OR opperator in non-boolean math notation

I'm trying to represent the boolean opperation OR in a regular formula, I am familiar with the boolean algebra notation, I came up with this (A+B)/(A+B) this works ...
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Can this Boolean expression be simplified any further?

I have simplified a Boolean expression to $$(\lnot a \land \lnot b \land \lnot c) \lor (a \land (b \lor c)).$$ Is there any way to simplify this further, e.g. using De Morgan's or anything?
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Proving that a given operator is universal.

I've got the next operator: $L(W,X,Y)=(W+Y)X'$ I need to prove whether the operator is universal, And if it isn't is: {$L,1$} or {$L,0$} are universal. I know that what i need to do is either ...
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Where did I go wrong with this Boolean simplification?

I am completely new to Boolean algebra, and I've tried to simplify this expression. All I did is tried to follow my lecturers methods, but I don't think it's right, and I have no idea how to do it. ...
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Do these equivalence classes have any special property!

Suppose that $\lambda$ be a measure on the interval $I=[0,1]$, and Let $\mathcal{N}$ be the family of null sets. It is known that "measure algebra $\mathcal{B}$ " is the algebra of all measurable ...
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My Boolean Expression Simplifications are correct or not?

1) I just want to know that my steps are correct or not? what are the missing steps. please help me. 2) Actually I can not simplify this is. So what are the missing steps at my trying path? please ...
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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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simplifying boolean algebra expression [duplicate]

A very urgent question : 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 : ...
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1answer
49 views

Boolean Expression Simplification

Dear all, I need help to simplify this Boolean algebra. Please give me a answer step by step. Thanks
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Boolean endomorphisms vs endofunctions on finite sets

I stumbled upon a funny fact: Let $\mathbf{Bool} = \{0, 1 \}$. For all functions $f: \mathbf{Bool} \to \mathbf{Bool}$ it is the case that $f^3 = f$. This got me excited and I was wondering whether ...
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How to give an assignment of boolean values such that this expression is evaluated to true?

Given the expression $E$, is there an assignment of boolean values ($true$ or $false$) that we can give to our variables such that this is evaluated to $true$? $E = (¬x + z + ¬v) · (¬v + w) ·(¬z + ...
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640 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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4answers
121 views

Boolean Algebra simplify minterms

I have this equation $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot \bar{C} + A \cdot B\cdot C$$ and need to simplify it. I have got as far as I can and spent a good 2 ...
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24 views

Demultiplexor Equation?

I am currently working on a project that requires a demultiplexor to be used. My problem is that I want to represent it in equation form, but no matter how much I try I cannot find the equation for a ...