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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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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0answers
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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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3answers
62 views

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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1answer
60 views

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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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
144 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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1answer
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 ...
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1answer
99 views

Inverse of a Boolean Function

Assume that I have a multi-output Boolean function $f(x_1,x_2,x_3,x_4) = (y_1,y_2,y_3,y_4)$. Is there a direct way of computing the inverse, that is, $g$ such that $g(y_1,y_2,y_3,y_4) = ...
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Simplify Sum of Products: $\;A'B'C' + A'B'C + ABC'$

How would you simplify the following sum of products expression using algebraic manipulations in boolean algebra? $$A'B'C' + A'B'C + ABC'$$
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1answer
88 views

Convert 4-sat to 3-sat

I want to know in general how can I convert $4-SAT$ to 3-SAT. And I have a specific case that if you can help me optimize it to 3-SAT it will be greate. I want to do this so I be able to use sat ...
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1answer
205 views

How to prove this equality using boolean algebra?

I have approximately no idea on how to solve the following problem, so any help would very much be appreciated: $$x' y'+ x y = (x y' + x' y)'$$ I can't figure out how to prove the equalities ...
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56 views

Boolean Algebra - Why is the result 1?

Given: = !(A * (!B + C)) + !(!B * !C) = !A + (B * !C) + !B + C Where: ! = NOT + = OR * = AND I'm having some trouble to why !A + (B * !C) + !B + C simplifies to 1? Can someone shed some light on ...
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0answers
245 views

Doubt in K-map rule (loops)

Is it possible to mark a loop in k-map as in second figure in above diagram (indicated in blue). According to my knowledge the first figure is correct. Is figure 3 or 4 correct? Note: Consider ...
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1answer
169 views

Cantor-Bernstein theorem for $\sigma$-complete Boolean algebras.

I am working on problem 7.28 from Jech's Set Theory: Let A and B be σ-complete Boolean algebras. Let a and b be elements of A and B respectively. If A is isomorphic to B$\upharpoonright$b and B is ...
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2answers
85 views

How to write this as a boolean expression?

How can I write the following sentence as boolean expression: $$ \text{If two sides of triangle are the same, then two opposite angles are the same} $$ I konw it should be something like this: $$ a = ...
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497 views

Convert Circuit SAT to 3-SAT

I am trying to convert Integer Factorization to $3-SAT$. So far I managed to convert it to Circuit SAT, but I don't know how to make the final step. This is how it look for 3*3 multiplication: ...
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1answer
38 views

Homework: Conjunctive Normal Form

The way I understand CNF is as an expression containing AND's of OR's. So an AND-GATE with 3 inputs (A, B and C) should just be A AND B AND C. But apparently this is incorrect. Any guidance would be ...
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66 views

Fields of sets in which, if the l.u.b. of a subset exists at all, it is the union of the subset

I am learning about boolean algebras and how they can be represented as fields of sets. Stone's representation theorem tells us that every boolean algebra is isomorphic to a field of sets. Consider an ...
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41 views

Prove This Bool Expression

Prove $x'z+xyz+xy'z=z$ can you show how you solve this using Boolean Algebra. My main problem is when I do this $xz (y + y') = 1 $ So $1$ times $x =$ ?
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1answer
24 views

Injectivity of Stone embedding

Let $\mathcal{B}$ be a Boolean algebra, $X$ the set of ultrafilters of $\mathcal{B}$ and $\sigma:\mathcal{B}\longrightarrow\mathcal{P}(X)$ the map sending $b\in\mathcal{B}$ to the set of ultrafilters ...
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2answers
83 views

A detail in the proof of Stone representation Theorem

Let $(\mathcal{B},\sqcap,\sqcup,\leq)$ be a Boolean algebra. Let $x,y\in\mathcal{B}$. I want to prove the following implication: $$x\sqcap y'\leq 0\Rightarrow x\leq y$$ where $y'$ is the complement ...
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637 views

Show that $ \{\lnot,\leftrightarrow\} $ is not functional complete

I have to prove that this set of logical operators is not functional complete - $$ \{\lnot,\leftrightarrow\} $$ i've tried implement this set by $ \{\rightarrow,\lor\} $ which is not functional ...
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1answer
210 views

Find the disjunctive normal form and then simplify

Let $f(x,y,z,w)=zw+z'w'+xy'z'w+xyz'w$ Disjunctive normal form $zw(x'+x)(y'+y)+z'w'(x'+x)(y'+y)+xy'z'w+xyz'w=(zwx'+zwx)(y'+y)+(z'w'x'+z'w'x)(y+y')+xy'z'w+xyz'w$ ...
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1answer
30 views

Proving at boolean algebra

Must prove that $$(X+Y )=X+(X.Y')$$ i tried a lot of ways, using logic things and expanding this things, but cant reach the Y. $$(X+Y )=(X+X).(X+Y')$$ Whats the possible prove to this?
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45 views

Equivalent definitons of atom in a Boolean Algebra

I want to show that the following conditions are equivalent for a nonzero element $a$ in a Boolean algebra $\mathcal{B}$: 1) for all $x\in\mathcal{B},a\leq x$ or $a\leq x'$ 2) for all ...
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1answer
33 views

how do you factor this boolean equation?

How do you factor this boolean equation $A'B'CD+AB'CD'+AB'C'D+ABCD$ I need help with where do I start from. What are the factors?
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1answer
47 views

In a boolean matrix, what does the $n$ in $M_{R^n}$ represent?

I'm now learning about binary relations. I stumbled upon this question in the book: Given $A = \{1,3,5,6\}$ and $R$ is a relation over $A$, whose matrix is defined by $$\begin{pmatrix} 0 ...
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1answer
338 views

Simplifying Sums of Product Expression obtained from 8-3 Priority Encoder (Computer Science)

I have an example for simplifying expressions in sums of product form, but I can't figure out which algebraic theorem was used to get rid of some of the variables: Step 1. (A'B'C'D'E'F'G) + ...
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173 views

Correct progression from DNF to CNF?

Trying to figure out how to transform this predicate from disjunctive normal form to conjunctive normal form (repost of an earlier question): $$( P \land Q ) \lor ( R \land S ) \lor ( P \land S )$$ ...
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Are there impossible boolean constructions?

I was reading about logic and I remember, for example: That with the binary $\mathtt{NAND}$ connector can be used to assemble all the other binary connectors - I already know that there are primitive ...
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1answer
46 views

Is this a valid re-write rule?

In my job (SQL developer) I frequently need to change search conditions (WHERE clauses, database constraints) from disjunctive normal form to conjunctive normal form (CNF). I confess I usually resort ...
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1answer
190 views

How to simplify the following expression through Boolean algebra

Disclaimer: This was a homework problem from the first assignment of the semester - the assignment has long since been graded. For the life of me I can't crack this one - I don't understand what I'm ...
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4answers
180 views

How is $((X\to Y)\to X)\to X$ a tautology?

$((X \rightarrow Y ) \rightarrow X) \rightarrow X$ converted to its disjunctive normal form is $X' + X$. Why/how does this show me why this formula a tautology?
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1answer
139 views

Convert $(X\lor Y)\land(W \lor Z)$ to disjunctive normal form

Using the distributive laws, I need to convert the formula $(X\lor Y )\land (W \lor Z)$ into disjunctive normal form. The answer needs to be equivalent to this formula by means of a truth table. Can ...
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2answers
42 views

Simplification of boolean algebra from “not s and p” to “not s”

I am trying to learn more about "Rules of Inference" and their application, but one thing always confuses me, and that is simplification "not s and p" to "not s". I have looked at some examples: ...
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1answer
72 views

Extending a Filter in a Well-Ordered Boolean Algebra to an Ultrafilter WITHOUT the Axiom of Choice

Hypothesis: Let $B$ be a well-ordered boolean algebra and let $F \subseteq B$ be a filter on $B$. Goal: Show that $F$ can be extended to an ultrafilter without the axiom of choice (or any equivalent ...
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1answer
141 views

Is every Boolean algebra a separative partial order?

A partially ordered set $\langle P,\leq\rangle$ is separative iff it satisfies the following condition: \[ \neg x\leq y\Rightarrow\exists z(z\leq x\wedge z\bot y) \] where: \[ x\bot y\iff\neg\exists ...
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1answer
233 views

Minimization of boolean function using Quine–McCluskey algorithm

I have a boolean row. It looked like this: Y = 0,1,0,1,1,0,0,1,1,0,1,0,1,1,0,0 Then I converted it to: f(x1,x2,x3,x4) = 0101 ∪ 1001 ∪ 1010 ∪ 1100 I divided it into groups: 0 | - ...
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118 views

How to minimize $\bar{A}.\bar{C}+\bar{A}.B+A.C$ further?

$\bar{A}.\bar{C}+\bar{A}.B+A.\bar{B}.C+B.C$ $=>\bar{A}.\bar{C}+\bar{A}.B+A.\bar{B}.C+\color{Orchid }{(A+\bar{A}) }.B.C$ ...
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1answer
196 views

Efficient division using binary math

I'm writing code for an FPGA and I need to divide a number by $1.024$. I could use floating and/or fixed point and instantiate a multiplier but I would like to see if I could do this multiplication ...
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1answer
145 views

Simplifying Boolean Algebra law

I've got a problem here that I could use help solving. I have simplified it to this point. Using Wolfram Alpha, I know it is still possible. My lecturer did it but I didn't catch all of it. It is ...
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1answer
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Boolean algebra generated by value sets of polynomials over $\mathbb{N}$

Update For each polynomial $P \in \mathbb{N}[X]$, let $S_P = \{ P(n) \mid n \in \mathbb{N}\}$. Does the Boolean algebra generated by the subsets $S_P$ of $\mathcal{P}(\mathbb{N})$ such that $P$ is ...
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1answer
2k views

Is it logically valid to prove DeMorgan's laws using the duality of boolean algebra?

I'm taking an introductory course in boolean algebra, and have been assigned the task of proving DeMorgan's Laws (so, disclaimer, this is homework). One line of reasoning that I came up with would be ...
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1answer
356 views

Boolean formula over 64 Boolean variables X

This question comes from this homework assignment from ECS20 at UC Davis. Chess is played on an 8 x 8 board. A knight placed on one square can move to any unoccupied square that is at a distance ...
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Prove the following with equivalence statements.

I need to prove the following statement with equivalence statements. $\exists x \in D,(P(x) \Rightarrow Q(x)) \ \text{is equivalent to} \ (\forall x \in D, P(x)) \Rightarrow (\exists x \in D, ...
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Boolean Algebra A + AB = A

Hi I have a question about the following algebra rule A + AB = A My textbook explains this as follows A + AB = A This rule can be proved as such: Step 1: Dustributive Law: A + AB = A*1 = A(1+B) ...
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166 views

Proving logical equivalence: $P \Leftrightarrow P \vee (P \wedge Q)$

I'm a first year CS student about to write his first term test and this question is part of our practice package. I have not been successful in writing a sequence of equivalences to justify this ...
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589 views

Disjunctive normal form expansion

I do not understand this at all. Find the sum-of-products expansions of these Boolean functions. $F(x, y, z) = x + y + z$ $F(x, y, z) = (x + z)y$ $F(x, y, z) = x$ $F(x, y, z) = x y$ ...