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 Logic using proofs

ABC' + C = AB + C I understand this using venn diagrams and intuition. However, I am not able to derive the proof for getting from one side to the other. It's probably very simple step that I keep ...
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Demonstrate AB+C(A+B)=AB+C(A⊕B)

Please help me demonstrate that AB+C(A+B)=AB+C(A'B+AB'). I've tried a couple of times but i always reach AB=2AB .
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Ideal generated by a set of polynomials $X^{a/b}$ where each monomial having $a$ and not having $b$

Let $$\mathcal R=\mathbb Z_2[x_1,\dots,x_n]/\langle x_1^2-x_1,\dots,x_n^2-x_n\rangle.$$ I want to learn ideal arithmetics to deal with polynomials of the forms such as Consider a set of ...
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Quick question on Bitwise operations

I have some questions for homework to do with Bitwise operations, now it's a simple task but it doesn't actually explain how to handle the questions which is why I'm asking here before I begin ...
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2answers
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Simplify $AC'+A'C+BCD'=AC'+A'C+ABD'$

How to prove that $$AC'+A'C+BCD'=AC'+A'C+ABD'$$ approch: a way to demonstrate is expressed in its canonical form. Any hint would be appreciated.
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How can I show a set B with 8 elements and two operations, such that the axioms of huntington for boolean algebra holds?

If It was about two members I would have choose B={0,1} with the operations: AND , OR And prove this. But how can I do this with 8 elements?
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Boolean algebra; what does <-> mean?

Expression :$$(p\rightarrow q)\leftrightarrow(\neg q\rightarrow \neg p)$$ What does the symbol $\leftrightarrow$ mean ? Please explain by drawing the truth table for this expression and also with ...
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1answer
39 views

Boolean functions that are not too far from all linear functions.

Suppose I have a Boolean function $ f:\mathbb{F}_2^n\rightarrow \mathbb{F}_2 $ which satisfies the following property: $$d(f, \ell)\leq 2^{n-1}\quad\forall\; \text{linear functions}\; \ell .$$ where ...
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Boolean Simplification $ABC' + BC'D' + BC + C'D$

I'd like to simplify this equation: $ABC' + BC'D' + BC + C'D$ prove it to $B + C'D$ My attempt is : $$\begin{align} &= ABC' + BC'D'(A+A') + BC + C'D\\ &= ABC' + ABC'D' + A'BC'D' + BC + C'D\\ ...
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22 views

Boolean expression explanation

Could someone explain how to get the following Boolean expression in its simplest form, I am having difficulties working it out step by step $$A+B+A*B$$
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1answer
23 views

Boolean functions- depth of generated function and info

I'm looking for a general book/link to information about boolean function (Function from to {0,1}), we've introduced them in a logic course but it seems we won't focus on them.
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Boolean Algebra - Prove XYZ + XYZ' + XY'Z + X'YZ = XY + XZ + YZ

Trying to prove $((X\land Y\land Z)\lor (X\land Y\land \lnot Z)\lor (X\land \lnot Y\land Z ) \lor (\lnot X\land Y\land Z)) \equiv ((X\land Y)\lor (X\land Z)\lor (Y\land Z))$ and I am a bit stuck. I ...
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4answers
35 views

Boolean Algebra - Xor simplification

I have a boolean equation: $e(g \oplus (g + b))$ and it is simplified to $e(\lnot g)b$. I do not see how this simplification is done. What i did was the following: $e(g \oplus (g + b)) --> ...
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1answer
30 views

Do we complement Boolean variables in the Dual?

The Principle of Duality states that starting with a Boolean expression, another Boolean expression can be obtained by : 1. Changing OR to AND 2. Changing AND to OR 3. Changing 0 to 1 4. Changing 1 ...
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Simplifying a Boolean algebra equation

I have a boolean algebra equation that i'm not able to simplify fully. \begin{align} &(c+ab)(d+b(a+c))\\ &(c+ab)(d+ba+bc)\\ &cd+ abc + bc^2+abd+a^2 b^2 + ab^2 c\\ &\text{using boolean ...
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1answer
20 views

Delove the truth for the three function same table

Given Boolean functions: $F(x,y,z)=x'.(y'+z')(x+y'), G(x,y,z)=x'.(z+yz')(x\oplus zy')$ Develop the truth table for the three function in the same table
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2answers
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Who is Petrick from Petrick's method?

I would like to ask your help. I think this is the best place for this. In my language -as well as English- I haven't found anything about Petrick yet. His method okay, but I would like to know about ...
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1answer
30 views

Boolean Algebra Problem [closed]

$ab+(ac)'+ab'c(ab+c) = 1$? how ?
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1answer
115 views

Existence of surjective homomorphism between Boolean algebras $\Lambda\subset\mathscr P(\mathscr B)\to\mathscr B$ (in ZF)

I am trying to prove the following theorem, due to Tarski according to W. A. J. Luxemburg on Reduced powers of the real number system and equivalents of the Hahn-Banach extension theorem: Given a ...
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34 views

Does this always evaluate to true?

This expression: $1 \lor (0 \land 1 \land 1 \land 1 \land 1 \lor 0)$ Regardless of how order of operations inside the parentheses are taken, which are ambiguous, the fact that it is and Or operation ...
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Order of Galois connections between two boolean lattices

Is the poset of Galois connections between two boolean lattices itself a boolean lattice? If not, does it hold for: complete boolean lattices? atomic boolean lattices? atomistic boolean lattices?
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Galois connections between boolean lattices - an alternative representation

Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) boolean lattices (with lattice operations denoted $\sqcup$ and $\sqcap$, bottom element $\bot$ and top element $\top$). I call a boolean funcoid a ...
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Two alleged counterexamples (about boolean algebras)

Trying to solve this question, I propose two possible counter-examples. Please help me to understand whether these cases are really counter-examples. Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) ...
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1answer
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More on a construction on two boolean lattices

Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) boolean lattices (with lattice operations denoted $\sqcup$ and $\sqcap$, bottom element $\bot$ and top element $\top$). I call a boolean funcoid a ...
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21 views

Boolean lattices vs boolean rings

Which kinds of theorems about boolean algebras are easier to prove with boolean rings (than with actual boolean lattices)? Give me at least one example, as an answer.
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Counting self dual functions

The dual of a function is defined as the same function with and/or operators exchanged .A function is self dual if dual of function and function itself are same.How many maximum self dual functions ...
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Simplifying a boolean expression without using sum-of-products

Yes this is for a school class, and no I'm not asking anyone to do my homework. This is from an example. I have a boolean algebra expression that I have to simplify. I also have the answer simplified ...
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1answer
29 views

Implement boolean function with OR-NAND and NOR-OR gates

This is the boolean function: F(A,B,C,D) = Σ (0,4,8,9,10,11,12,14) and so after using a K-map to minimize it, I came out with F(A,B,C,D) = C'D' + AB' + AD'. Now the other two parts of the problem were ...
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Boolean Algebra, simplify expression

I've been trying to simply this expression but all I have managed to do is get rid of a D'. I assume there must be more I can do but I can't find out what I'm supposed to do to it. If you can help me ...
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1answer
22 views

Boolean Algebra, Proving expression

I've been struggling to prove this expression literally all day please help. A'B + A'C + B'C = A'B + B'C
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Simplify expression in Boolean algebra

In Boolean algebra, I need to prove that $AB+AB'C+BC'=AC+BC'$ and $(ABC)'(A+B+C)=A'B'C'$ Are both the questions correct?
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1answer
38 views

Boolean Algebra- Simplification

I'm attempting to simplify this and don't know if I'm doing it right. This is the problem: (a+b+c')(a'b'+c) Attempted solution: ...
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Negation of XOR

I feel pretty confident with expanding an XOR, but when it is negated, it throws me for a loop a bit. The problem I am trying to prove: $$\overline{x_1 \bigoplus x_2} \bigoplus x_3 = ...
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Write PQ' in the form A'+B

I'm trying to find the negation of the sentence "All domestic cars are good". The sentence can be rewritten as "If a car is domestic, it is good". if P then Q is the boolean expression - P'+Q. The ...
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Simplifying bitwise expressions

Using various methods it is possible to simply boolean expressions consisting of boolean operators and binary variables. In programming languages another closely related set of operators exists: ...
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1answer
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Prove the following boolean identity using Consensus theorem.

I have been trying to prove it for last 4 hours but couldn't find a solution. Please help me. $$(A+B')(B+C')(C+D')(D+A')=(A'+B)(B'+C)(C'+D)(D'+A)$$ I solved and got the following answer. ...
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1answer
48 views

If $*$ a functionally complete logical operator then $tautology *tautology $ is contradiction.

Would anyone please give me a hint to prove that if $*$ is a functionally complete binary connective and $@$ is a symbol for tautology, we must always have $ @*@$ is equivalent to contradiction (I ...
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1answer
62 views

Free product of the powerset algebras of group orbits. Interpretation

I am trying to interpretate the following sentence in the context of measures on groups and algebras, from J. Pawlikowski on "The Hahn-Banach theorem implies the Banach-Tarski paradox" Let $F$ be ...
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Resolution of a Boolean Function

I have to solve this simple boolean function : $$f_1 * f_2 = (x_1 + x_2) * (!x_1 + x_3)$$ The solution is : $x_1*x_3 + !x_1*x_2$ Can anyone make a step by step solution because after getting : ...
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1answer
37 views

All possible set operations between sets

I was looking for some sort of generalization on set operations between different sets, and how that number of operations increases as the number of set increase as well. It can also be thought as the ...
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4answers
36 views

Simplify the boolean function below by using algebra laws.

I've been stuck on this question for some time, if anyone happens to solve it please explain step by step. $$(A +B ) \times ( A' + C ) \times ( B + C )$$
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1answer
55 views

Intuition for power-set structure of finite Boolean rings

A course I am taking has started to introduce Boolean rings: rings where every element is idempotent. It was proved that every finite Boolean ring $R$ is isomorphic to a power set ring $\wp (S)$ for ...
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Why are there two different notations for negation in boolean logic?

For the boolean variable $x$, there are two notations for its negation: $\neg x$ and $\bar x$. So why are there two different notations?
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Karnaugh map and Circuit of a full adder

I have the following task: The addition can be implemented by the rules 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10. Full addition requires carry-in and ...
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Boolean equation

$$\text{Solve for}\space{x, y}$$ $${a_1, a_2, a_3, a_4, b_1, b_2} \; \text{ - variables}$$ $$\left\{ \begin{aligned} {a_1}\&x \oplus {a_2}\&y &= {b_1} \\ {a_3}\&x \oplus {a_4}\&y ...
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Is there a proof for the FOIL method in Boolean algebra?

The FOIL method is the special case of multiplying algebraic expressions using the distributive law and is shown here: What does the proof for this look like using Boolean algebra?
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m-bit parallel adder needed using full adder

How many full adder is needed to construct a m-bit parallel adder? I have construct a 4-bitparallel adder with 4 full adders. but can the number be reduced?
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Need help on clarification on a boolean algebra/logic gate question.

The question asked on my homework. I have a question on my home work that is confusing me. I went through and made a truth table and found the all of the values corresponding to the minterms that ...
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1answer
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Can you simplify this Boolean expression any farther?

I was working through a problem for a Computer Engineering course and i was given this logic function F(A,B,C,D) = ~A~BC~D + ~AB~C~D + ~ABC~D + ABC~D After ...
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1answer
26 views

An intuition connected with Heyting implication

Suppose $L$ is a bounded lattice and let $\Rightarrow$ be its Heyting implication, i.e. the operation defined in the standard way: $x\Rightarrow y$ is the largest object of the set $\{u\in L\mid ...