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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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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complete set of connectives [on hold]

How can I prove that $\{f,XOR\}$ is a complete set of connectives which : f = a'*(b XOR C) (b XOR c = b'c+b c') I try so hard to prove it , but i didn't succeed i need help!! I can get a+b and i ...
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1answer
82 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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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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2answers
17 views

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
16 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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2answers
4k views

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

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 = ...
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1answer
21 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
41 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
363 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
5 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
26 views

abstract boolean algebras prove De morgans law x'+y'=x'y' [closed]

Im stuck on this one, Im not sure how to apply the boolean properties / laws. Im supposed to prove and not invoke the duality principle or a truth table. Prove De morgans law x'+y'=x'y'
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1answer
24 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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1answer
341 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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2answers
99 views

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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45 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
22 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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54 views

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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2answers
64 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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1answer
15 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
39 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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17 views

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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20 views

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 ...
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32 views

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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2answers
495 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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44 views

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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33 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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20 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 ...
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Question on Boolean Algebra - Atomicity and Completeness

I am trying to solve some Boolean Algebra exercises from the book of Mathematical Logic by Cori and Lascar. I am having some problem in solving a question. Please help me. Thnx in advance. I was ...
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59 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
11 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
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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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What is the algorithm to add 2 binary with boolean operations? [closed]

What is the algorithm to add up 2 binary numbers when the basis is {negation, conjunction, disjunction} in linear time? Also the program needs to be linear as well, meaning there can only be ...
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1answer
653 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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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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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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Probability of boolean function

I have a boolean function $F(x_1,...,x_n)$ given in disjunctive normal form. $x_1,...,x_n$ are independent random boolean variables following Bernoulli distribution, i.e. $x_i = True$ with probability ...
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631 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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How does it evaluate A XOR B XOR C?

I am trying to solve the following combination, ...
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1answer
24 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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1answer
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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 ...
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43 views

Boolean Algebra: Simplifying $\;xyz + x'y + xyz'$

Given the following expression: $xyz + x'y + xyz'\,$ where ($'$) means complement, I tried to simplify it by first factoring out a y so I would get $\;y(xz + x' + xz').\,$ At this point, it appears ...
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Basic Boolean Algebra Multiplication Question

I have the following term $$ t1: \overline {\overline{x1x2\Leftarrow\Rightarrow x1x3}\Leftarrow\Rightarrow x2x3} $$ which I already converted to this: $$ t2: ((x1x2\overline{x1x3} + ...
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1answer
176 views

What is the number of self dual boolean functions?

The dual of a Boolean function $F(x_1,x_2 \dots x_n,+,\bullet)$, written as $F^D$, is the same expression as that of $F$ with $+$ and $\bullet$ swapped. $F$ is said to be self-dual if $F=F^D$. What is ...
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1answer
16 views

Equivalent form of biconditional

I'm reading How to Prove It: A Structured Approach (Velleman) Second Ed. Doing all the end of chapter exercises for chapter 1 and having trouble on problem 5a which reads Show that $P ...
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1answer
26 views

The proof of that a → b is equivalent to ¬b → ¬a using algebraic identities by ArsDigita

I'm noob practicing with discrete math problems, and not sure if the solution ArsDigita provided for this one is correct or not: Prove that a → b is equivalent to ¬b → ¬a using algebraic identities. ...