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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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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840 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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1answer
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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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1answer
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Count of boolean functions satisfying a condition [closed]

How to find the number of boolean functions of n variables with the given property: If $f(\alpha) = 1$, where $\alpha \in J^2_n$, then $\forall \beta \in J^2_n : w(\beta) \ge w(\alpha) \implies ...
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Boolean Expression Problem [closed]

X or (not X and Y) or(not X and not Y)
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42 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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2answers
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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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2answers
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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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1answer
30 views

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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2answers
64 views

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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2answers
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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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1answer
25 views

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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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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1answer
29 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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0answers
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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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3answers
106 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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2answers
518 views

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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1answer
679 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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2answers
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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
19 views

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
26 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
43 views

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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2answers
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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1answer
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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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26 views

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 ...
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1answer
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Boolean expressions from multiplication to addition and vice-versa

I am trying to change these Boolean expressions into expressions that do not use multiplication. Bolds indicate complements. a) abc b) (ab +c)d And these to ones that do not use addition. c) a + b ...
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2answers
33 views

Proving that a set with a quaternary logical connective is functionally incomplete (i.e. inadequate)

I am stucked at trying to prove that the set $\{N\}$ of one logical connective is inadequate where $N$ is a quaternary connective that is defined as follows: $N(w,x,y,z)=((x\land y)\land(w\lor z))$ ...
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Boolean Algebra - proof without associativity?

I would like to prove the following: $(x\cdot y) + (\overline{x} + \overline{y}) = 1$ without the Associativity Property. I can't seem to do this algebraically (without truth tables).
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Does disjunction of two Boolean algebra cuts always produce their ideal sum?

Let $(B, 0, 1, \leq, \wedge, \vee, \neg)$ be a Boolean algebra. For a subset $A \subseteq B,$ denote by $L(A) = \{l \in B \mid (\forall a\in A) \, l \leq a\}$ the set of all lower bounds of $A,$ and ...
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1answer
31 views

How to simplify Boolean Expression $\bar B + \bar C (B + A)$

I trying to figure out how $ \bar B + \bar C (B + A)$ simplifies to $ \bar B + \bar C$.
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1answer
19 views

How do I input this Boolean Expression into a K map?

Determine the minimum SOP, sum of products expression using K-Map F(A,B,C,D,E) = (A’ + B + C’ + D + E’)(A’ + C’ + D + E )(A’ + C’ + E )AC’ Do i have to actually simplify it first by multiplying ...
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how to convert XOR to DNF form?

If I have the expression $X$ xor $Y$ , how do I convert it to DNF form? and $X$ implies $Y$? I only got to the point where I make the truth table for the functions, but after that I could not apply ...
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What are The applications of Fast Walsh–Hadamard Transform.

There is a problem requiring the expect value of the intersection of two random subsets selected from a universal set, with the values and the probabilities of subsets given. My friend said it could ...
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1answer
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xor-ing vectors

This question might be wrong on mathematics, but I don't know where else to put it. I have a given equation, and there is one calculation step, that I don't understand. I thought, I have to xor ...
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Does a logical matrix representing sets have a name or special properties?

Imagine a collection of separate objects and several sets. These sets can be represented using a logical matrix. $M = \begin{bmatrix} 1 & 1 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 ...
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What is the algorithm to add binary numbers with boolean operations? [closed]

What is the algorithm to add up two binary numbers using only boolean operations (negation, conjunction, disjunction) in linear time? Also the program flow needs to be "linear" as well, meaning there ...
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1answer
29 views

Closedness of $\{ x \in 2^A : x(\neg p) = \neg x(p) \}$ for a Boolean algebra $A$ and $p \in A$

I'm reading Matthew Dirk's The Stone Representation Theorem for Boolean Algebras, and am trying to follow the proof of Proposition 3.4 on p.6: Proposition 3.4. Let $A$ be a Boolean algebra, and ...
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1answer
17 views

Finding Prime Implicants and Essential Prime Implicants for Boolean Functions

I am trying to solve a EE problem and am unsure whether I doing it correctly. The problem is: Find all the prime implicants for the following Boolean functions, and determine which are essential: ...
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1answer
29 views

Boolean algebras and rings

I know that M. H. Stone proved that there is a bijection between boolean algebras and boolean rings. The bijection I know is the following: to any given Boolen algebra $(L,\, \vee, \wedge)$ we ...
2
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1answer
68 views

Explain why the description defines a Boolean Algebra

This is the exercise: Let $A = \{a,b\}$ and list the four elements of the power set $\mathcal P(A)$. We consider the operations $+$ to be $\cup$, $\cdot$ to be $\cap$, and complement to be set ...
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3answers
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Solving this logical puzzle by resolution doesn't work for me

In this document there is a logical puzzle: If the unicorn is mythical, then it is immortal. If the unicorn is not mythical, then it is a mortal mammal. If the unicorn is either immortal or a ...
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4answers
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How to prove this tautology using equivalences?

I am trying to prove that the following is a tautology: $(A \implies (B \implies C)) \implies ((A \implies (C \implies D)) \implies (A \implies (B \implies D)))$ To make progress, I thought I'd ...
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Do 'sum-of-products' and 'product-of-sums' represent the same function?

Do 'sum-of-products' and 'product-of-sums' represent the same function? Does it have be the same expression or not? In case it is different, what does it mean? Context: I've just made a Karnaugh map ...
2
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1answer
60 views

Proving that a set with a ternary logical connective is functionally incomplete (i.e. inadequate)

I am stucked at trying to prove that the set $\{\lnot ,G\}$ of logical connectives is inadequate where $G$ is a ternary connective that gives $T$ (True) if most of its arguments are $T$. For example: ...
2
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2answers
76 views

Show that every boolean function with 3 variables can be represented with maximum number of 10 gates

I need to show that every Boolean function with 3 variables can be represented with maximum number of 10 gates, limited to the following: AND(2 ins), OR(2 ins), NOT(1 in). I tried to find Boolean ...
3
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3answers
71 views

What's the relationship between continuity property of Lebesgue measure and continuity on a metric space?

This is a topic from Lebesgue measure in $\textit {Carothers' Real Analysis}$: I know how to prove Theorem 16.23. However, I can not figure out why he names this property as continuity? Besides ...