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

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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42 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
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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
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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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2answers
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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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53 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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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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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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1answer
21 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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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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51 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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36 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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1answer
23 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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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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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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1answer
106 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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1answer
25 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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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
20 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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2answers
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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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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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26 views

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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2answers
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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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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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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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3answers
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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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1answer
18 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
31 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
28 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. ...
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Monotonic operators in classical logic

Which means monotony for a logical operator, and affinity, in propositional calculus affinity..., here on wiki do not quite understand!!
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Simple Boolean Algebra Question

I have the following term in front of me: $$(AB+AC+\overline BC+B\overline C)*(A+\overline B+C+D)$$ and just need to multiply the whole thing which should result in this: ...
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Is the algebra of these circuits valid?

I drew these circuits when I was studying Boolean Algebra. Is the algebra of these circuits valid?
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proof that k-maps give the most minimized result?

how can i prove that a k-map for n variables gives the most simplified representation of a Boolean function ? (by simplified i mean we cannot eliminate another variable)
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1answer
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Understanding comparability in a partially ordered sets with Hasse diagrams

I'm doing a section on directed graphs and Hasse diagrams right now on partially ordered sets, and I'm trying to understand what it means for two elements in a Hasse diagram to be comparable. For ...
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1answer
26 views

Convert boolean expression to pos then nor only

I'm trying to convert a + xb + xyz to POS then to nor only. First I got, a'(x' + b')(x' + y' + z') by using the duality rule but then I get confused after that. Thanks.
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Find boolean function

Given $\mathbb{B} = \{true, false\}$, and function $f: \mathbb{B} \times \mathbb{B} \times \mathbb{B} \to \mathbb{B}, f(a,b,c) = a \land b \lor c,~ \forall a,b,c \in \mathbb{B}$. I want to find a ...
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Solving equation set with boolean operators and very specific format

I have to write a program to solve a set of equations like the following (+ is XOR and * is ...
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37 views

Expanding brackets in a logical expression.

So I have a logical expression, which I need to draw a Karnaugh map for. The expression is: $r = (\overline x+\overline z+\overline y)(\overline x\overline y\overline z+\overline x y) $ What would ...
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1answer
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Obtain the Boolean expression from the given circuit diagram

Currently having trouble understanding how to write out the boolean expression up to the exclusive or gate. Up to the third NAND gate I solved it to be AB+CD. But I get stumped on how to write out ...
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96 views

Is it possible to convert this expression into a NAND GATE Circuit?

I am trying to construct a logic circuit for the expression (NOT Q & P) OR R - using only NAND gates. I have tried this, can someone confirm if what I have done is correct? if not what do i need ...
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Coset state for non-abelian hidden subgroup problem

On page 14 of his classic paper, Quantum factoring, discrete logarithms and the hidden subgroup problem, Jozsa introduced a function $f$ for the non-abelian hidden subgroup representation of the graph ...
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Proof of the following statement?

For one of the inclass problems, we had to prove the following statment using Properties of Boolean Algebra: xyz + x'y'z + x'yz + xyz' + x'y'z' = xy + yz + x'y' ...
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1answer
46 views

Gelfand duality restricted to the category of Stone spaces

It is well known that the category of unital commutative $C^\ast$-algebras is dually equivalent to the category $\mathbf{KHaus}$ of compact Hausdorff spaces. I have found that restricting the ...
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A discrepancy in the total number of conjunctive normal forms and the number of distinct boolean functions.

Consider a set of truth literals $C$. The set $\{\text T, \text F\}^{\mathcal{P}(C)}$ is the set of all boolean functions over all subsets of $C$. This comes from the notation ...
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Product of binary Boolean operators

I'm interested in the set $\mathcal{P}_N$ of boolean functions of boolean variables $p_1, p_2, \ldots, p_N$ that can be written as products of operators of 2 variables only: $$ \phi(p_1, \ldots, p_N) ...
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Atoms in a Boolean algebra

I am trying to understand the concept of an atom in a Boolean algebra. To fix the ideas, let $X=\{a,b,c\}$ be a set, and $\mathcal{A}=\{\emptyset,\{a\},\{b,c\},X\}$ be one of the five possible ...
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Dual formula in propositional logic

There's something I don't understand in my course on propositional logic. In the case of x being a variable, the definition of its dual is x* = x. Right. However, further in the course, there's a ...
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What is the symbol you'd use for Boolean results?

What I mean is that $\mathbb{CRZ}$ etc. are used for different classes of numbers, allowing me to do stuff like this: $$f:\mathbb{R}\to\mathbb{R}$$ $$f:x\mapsto 3x$$ But say I have an expression ...
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Performing arithmetical operations (with binary numbers) using propositional logic

Clarifying some terms. By arithmetical operations I mean the four basic operations of addition, subtraction, multiplication and division. By binary numbers I mean numbers in the binary system. By ...
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How can I simplify this expression using the Consensus Theorem?

I'm doing some Boolean Algebra and there's this problem that I’m stuck with: Simplify $W'Y'Z + W'XZ + XYZ + WXY + WYZ'$ using the Consensus Theorem. My Attempt: $W(XY) + W'(XZ) + XYZ = WXY + ...