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: 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
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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
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
27 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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32 views

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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1answer
28 views

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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26 views

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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4 views

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
25 views

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
24 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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26 views

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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20 views

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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0answers
18 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
37 views

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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1answer
71 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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0answers
28 views

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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1answer
16 views

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
32 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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1answer
27 views

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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0answers
49 views

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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1answer
31 views

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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34 views

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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34 views

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 + ...
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1answer
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How to reduce a Boolean Algebra expression/function

I need to reduce this expression: $$F(A,B,C,D) = A'B'C'D + A'B'CD + A'BC'D + A'BCD' + AB'C'D + ABC'D' + ABCD'$$ I also have the following solution: \begin{align*} &= \bar A \bar B D + ...
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1answer
25 views

Exercise 1.9 in Rotman's homological algebra: ideals in boolean rings

We consider the Boolean ring $\mathcal{B}X$ of subsets of $X$, with the operations of symmetric difference as addition and intersection as multiplication. One direction of part iii of the exercise ...
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3answers
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Boolean simplification A'B'C' + A'BC + ABC'

Gentlemen I need a hint to simply this expression since I'm quite rusty in my boolean algebra. A'B'C' + A'BC + ABC' I however have made thus far ...
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1answer
110 views

Function which essentially depends on all of its variables.

I'm interested in the proof of the following theorem: Let the boolean function $f(x_1,x_2,..,x_n)$, depend essentially on all of its $n$ variables, $n ≥ 2$. Then there is an index $i$ and an ...
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4answers
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Can this set of rules perform all Boolean operations?

I never worked in this field before, I just thought about this set of rules and never saw something similar before. I apologise if I don't use the right mathematical vocabulary for my question. ...
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0answers
19 views

Characteristic polynomial of a 3-bit LFSR

So I am trying to solve part d of this question and using $IX - T$ I get the expression $(x+1) \oplus (X^2 + x + 1)$. I cannot solve it further. I am supposed to give a single polynomial. Can anyone ...
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Coproducts and pushouts of Boolean algebras and Heyting algebras

I am having trouble of find a reference explaining how to compute coproduts and pushouts in the category of Boolean algebras and in the category of Heyting algebras. To be precise I am looking for ...
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1answer
24 views

How to prove this equality?

I would like to prove the following using boolean algebra and not karnaugh maps but I'm stuck: CD' + CDAB' + C'D'AB' = CD' + CAB' + D'AB'
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1answer
12 views

Boolean Algebra Simplification

Can someone show me the steps of simplification for this Boolean expression? (!A!B!CD) + (!AB!C!D) + (!AB!CD) + (!ABCD) + (A!B!CD) + (ABCD)
2
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1answer
49 views

regular open (Boolean) algebra is complete

To prove that regular open (Boolean) algebra is complete, I tried to show following claim, but I couldn't. I saw this statement in Kunen's 'Set Theory' p.64 but in other books what I checked, ...
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2answers
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Definitions of Boolean algebras

One definition I find of a Boolean algebra in the book that I am following (V. Manca, Logica matematica, 'matematical logic') is determined by the binary operations $\land$ and $\lor$ and the unary ...
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1answer
60 views

Does $x(y+z)$ simplify to two variables in Boolean Algebra?

Question from the title. I'm just starting with Boolean algebra and my first set of exercises contains multiple problems which simplify to a variant of this. Am I "done" these problems, or can I still ...
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1answer
20 views

Is it possible to check if this function is associative without checking all the cases?

Given a boolean function with the following table: $$\begin{matrix} {A}&{B}&{out}\\ {0}&{0}&{0}\\ {0}&{1}&{0}\\ {1}&{0}&{1}\\ {1}&{1}&{0} \end{matrix}$$ ...
5
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1answer
152 views

Mixed Boolean Arithmetic Identity

I'm trying to prove/derive the equivalence of the following formula: $$ x*y = (x \land y) * (x \lor y) + (x \land \neg y) * (\neg x \land y) $$ whereas $(\land, \lor, \neg)$ correspond to bitwise ...
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1answer
42 views

Repeated XOR operations

Suppose you have a list of truth values with $2^k$ elements for any natural number $k$. If the first element of this list is denoted as $L(1)$, then we can come up with a new list by performing the ...
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11 views

Drawing a precondition decomposition diagram to prove mutual exclusivity

I have a question on my exam papers relating to proving mutual exclusivity by composing a precondition decomposition diagram. The problem is I'm not sure how to actually construct the diagram. I've ...
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2answers
99 views

Prove that if a and b are positive real numbers, then a + b $\geq$ ab

As the title states, the question is: Prove that if a and b are positive real numbers, then $a + b \geq ab$ For this proof, I'm supposed to prove by contrapositive. So, I get this as a general ...
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1answer
44 views

Is there any $\sigma$-algebra where its elements are equal to a finite disjoint union of generators?

Let $X$ be a set and $\mathcal{B}$ be a family of subsets of $X$. Let $\Sigma$ be the smallest $\sigma$-algebra that contains all elements of $\mathcal{B}.$ Under which assumptions it holds that for ...
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2answers
66 views

Proving equivalency using boolean algebra laws of logic

I have a question on my exam papers relating to proving equivalences using the laws of logic, but I'm not sure how to work it out as I don't have the solution paper. Can someone explain to me the ...
0
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1answer
32 views

Simplifying Simple Boolean XOR Expression (!AB + A!B)

I am trying to simplify the 5 gate XOR from a A!B + !AB expression to a (A + B)!(A + B) implementation. How can I convert ...
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1answer
20 views

Find the numbers by XoR

I have 6 numbers M1, M2 and M3 and E1, E2 and E3 such that M1 xor M2 = E1 xor E2 M2 xor M3 = E2 xor E3 ...
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1answer
34 views

Triple XoR - Find relation between the numbers.

I have a = b^c; b = a^c; Is it possible to eliminate c and find a relation between a and b? I have 3 different ...
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1answer
31 views

Solving a system of xor equations?

How can I solve the following system of xor equations? k0 ⊕ k2 ⊕ k3 = 0011 k0 ⊕ k2 ⊕ k4 = 1010 k0 ⊕ k1 ⊕ k2 ⊕ k3 = 0110 How can I solve this system to know the ...
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1answer
34 views

a bit complicated boolean simplification

I'm trying to simplify the following boolean expression: [(A' (C+D)')'] (A) + ( B (DC) + (D'C') + A + CB' What I got is A + (C+D) + B [(DC) + (D'C')] + A + CB' A(A+C) + D + B[1] + A +CD' A + D ...
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2answers
37 views

What's the name of this law in Boolean algebra?

I forgot the name of a law in Boolean algebra, and I can't think of how to ask this question to a search engine. It's the law that states that the disjunction of a variable with the conjunction of its ...