# Tagged Questions

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 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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### 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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### 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 AC'...
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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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### Number of Linear boolean-functions [closed]

How many linear boolean functions are there, if we have n variable?
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### Translating English to symbolic logic

(Question prompt) The domain of discourse in this problem is the set of students and teachers at a school. Define the following predicates: • E(x, y): x has sent a letter to y. • P(x): x is a ...
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### Prove that a boolean function using only $\vee$ and $\wedge$ must attain the value $1$ at least once

Please give me feedback on this Prove that a boolean function constructed only by using $\vee$ and $\wedge$ (without using $\sim$ ) must attain the value $1$ at least once.
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### $\sim p$ ,$\sim\sim\sim\sim\sim p$ , and $\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim p$ . Which of these are equal?

I made an attempt on this question. Please guide me if its wrong. Consider the following boolean fuctions: $\sim p$ ,$\sim\sim\sim\sim\sim p$ , and \$\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim ...
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### How to find/generate a 6 variable Bent Function?

I want to find a Bent Function with 6 variables. I read some papers about how to generate Boolean Functions, but I don't want to implement an algorithm from zero just to find one function. It is also ...
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### Boolean Algebra: a+a'b = a+ab = a?

a(a'+b) = aa'+a'b = a'b (aa' = 0 in any case) a+a'b = 1a + a(a'+b) = a(1+a'+b) = a a+ab = a(a+b) = a => a+a'b = a+ab However when I use truth table to compare the result of a+a'b to a+ab when a = ...
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### Essential Prime Implicants and Minterm Expressions

I have an exam for a university course shortly, and upon reviewing one of my assignments I have come to realize that I don't understand why I have lost marks/how to do a couple of questions. Hopefully ...
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### 3 input XOR gate

I am just beginning in computer engineering and need help with a problem. I have to implement a circuit following the boolean equation A XOR B XOR C, however the XOR gates I am using only have two ...
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### How does it evaluate A XOR B XOR C?

I am trying to solve the following combination, ...