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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1answer
18 views

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
152 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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26 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 AC'...
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1answer
19 views

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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2answers
383 views

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
31 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
543 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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41 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 ...
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1answer
108 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
137 views

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
91 views

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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1answer
188 views

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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1answer
250 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: ...
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2answers
97 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 ...
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3answers
87 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 ...
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1answer
26 views

Number of Linear boolean-functions [closed]

How many linear boolean functions are there, if we have n variable?
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97 views

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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1answer
45 views

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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1answer
93 views

$\sim p$ ,$\sim\sim\sim\sim\sim p$ , 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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1answer
131 views

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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2answers
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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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1answer
102 views

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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1answer
135 views

karnaugh map simplification

I really wonder why my method is wrong. Could you explain step-by-step and why my methods wrong. Drawings includes just one time isn't it enough for simplification ? First boolen expression: $$ F = A'...
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1answer
43 views

How would one solve this boolean algebraic equation?

During software testing I needed to find at least one solution for this: (a or (b and c)) != ((a or b) and c) Where all variables are boolean. I can (and did) ...
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3answers
45 views

Prove that $x+(\overline{x}\cdot\overline{y})=x+\overline{y}$

Prove that $x+(\overline{x}\cdot\overline{y})=x+\overline{y}$ The values of both these boolean functions show that these 2 are equivalent. ...
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1answer
468 views

Cardinality of the set of ultrafilters on an infinite Boolean algebra

Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?
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1answer
13 views

how to simplify (x+y')X(x+z')?

Hi this is for a Discrete Math test I have today. I can barely understand the simplification of boolean expressions. Can anyone show me if the (x+y')X(x+z') can be simplified further, what are the ...
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1answer
65 views

Boolean Expression Simplification XOR

I have been trying to express XOR in terms of just the '&' and '~' operators. I have managed to get the original XOR definition (~x & y) | (x & ~y) down to ~(x & y) & ~(~x & ~...
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422 views

Boolean Simplification

I'm having some trouble getting a handle with this course. We are starting Boolean algebra and my professor wants us simplify the following: (AB)'+(A'+B')'= (AB)'+BC+A'B'C'= I am assuming the "()" ...
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How to find the minimum expression(s) of a set of fixed-width bit fields?

If we define $x_1 x_2 \cdots x_n$ as a bit field of width $n$, and each element $x_i$ may be $0$, $1$, or wildcard $*$. A set of 4-width bit fields $\{0000, 0001, 0100, 0101\}$ can be aggregated ...
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1answer
64 views

Boolean Algebra: Is this equality or inequality?

Consider: $$xy + x'y' + yz = xy + x'y' +x'z$$ Is this equality true? I know I could a truth-table but I prefer doing it algebraically. I think there's something tricky here (Like adding a term, ...
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1answer
145 views

simplifying boolean expression in maxterm

Should I expand the equation to simplify? Π(1,4,5,6). It means $$ F = (A + B + C')(A' + B + C)(A' + B + C')(A' + B' + C) $$ I have expanded and found $$ = ( C' + AC + A'B)(A' + BC + B'C') $$ I haven't ...
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Boolean Simplification of AB + A'+B'

Is there any way to simplify this function? Or is this the simplest equation? : AB + A'+B'
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53 views

An effective way to find missing minterms

I've been messing with logic formulas lately and there was one thing that was often causing me headache. I'll describe it briefly. When using Quine-McClausky's algorithm for finding MDNF and MCNF, I ...
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1answer
139 views

How many binary bit strings of length 32 are there

How many binary bit strings of length 32 are there? I think I know the answer but I'm not sure...wouldn't it just be $2^5$ ?
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Simplifying $(\neg x\land \neg y \land \neg z) \lor (\neg x\land \neg y \land z) \lor (x\land \neg y \land z) \lor ( x\land y \land z)$

I'm looking at this logical formula: $(\neg x\land \neg y \land \neg z) \lor (\neg x\land \neg y \land z) \lor (x\land \neg y \land z) \lor ( x\land y \land z)$ Asked to simplify it as much as ...
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Boolean Algebra - reducing a function

Let $$f(w,x,y,z) = w'x'y'z' + w'x'yz' + wx'yz'$$ How can you reduce it to: $$x'z'(w' +y)$$ Thanks!
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A Criterion For a Set To Have all the Atoms of a Boolean Algebra

Let $\Omega$ be any set and let $\mathcal A$ be an algebra of sets in $\Omega$. An element $E\in \mathcal A$ is said to be an atom if there is no non-empty element $A\in \mathcal A$ such that $A\...
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1answer
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Boolean algebra proof - I don't know why this is valid!

So this is the answer proof I was given, I'm stumped by the final application of the Idempotent law (where does that 1 come from!?) As I understood it a 0 or 1 can only come from a combination of A ...
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How many ways are there to sit 4 people from a group of 10 people around a circular table where two sittings are considered

How many ways are there to sit 4 people from a group of 10 people around a circular table where two sittings are considered the same when everyone has the same immediate left and immediate right ...
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1answer
56 views

How to prove an equality in a Lindenbaum-Tarski algebra?

Let $\mathscr{L}'= \mathscr{L}\cup \mathscr{C}$ be an extension of the language $\mathscr{L}$ with a new infinite set of constants $\mathscr{C}$, and $T$ be an $\mathscr{L}$ theory. I wish to show ...
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30 views

Each Element of an Algebra can be Partitioned into “Atoms”

Let $\Omega$ be any set and let $\mathcal A$ be an algebra of sets in $\Omega$. An element $E\in \mathcal A$ is said to be an atom if there is no non-empty element $A\in \mathcal A$ such that $A\...
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444 views

What is the difference between Boolean logic and propositional logic?

As far as I can see, they only employ different symbols but they operate in the same way. Am I missing something? I wanted to write "Boolean logic" in the tag box but a message came up saying that if ...
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2answers
23 views

simplifying boolean expression in minterm

i am trying to simply the equation and stuck. Sum symbol(2,4,6,7). It means $$ F = A'BC' + AB'C' + ABC' + ABC $$ $$ = A'BC' + AB'C' + AB(C' + C) $$ $$ = A'BC' + AB'C' + AB $$ After the last equation ...
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1answer
46 views

expanding boolean expression as maxterm

$$ F = A + B'C $$ The expression has bothered. I've tried to expand the expression in maxterm, however, I'm stuck on the $B'C $ part. My approach is like this $$ = A + (B'B) + (C'C) + B'C $$ $$ = (A + ...
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3answers
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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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11k views

How does it evaluate A XOR B XOR C?

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

Reference for the fact: elements as union of atoms in a Atomic Boolean lattice [closed]

I need a reference to a book with the following statement: "In a Atomic Boolean Lattice every element is the union of the atoms under lie it". Does not matter if it is presented as a exercise.
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130 views

Consensus Theorem: Legal to use redundant terms to find more redundant terms?

When using the Consensus Theorem in Boolean algebra to minimize an expression, is it a legal move to find and add a redundant term to the expression and then use that term to find more redundant terms ...