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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Arrow's Impossibility Theorem Using Boolean Algebra

I am currently working on a research project which involves using Boolean matrices for the proof of Arrow's Impossibility Theorem and various other lemmas and results related to quasi ordered sets. In ...
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Boolean Least Squares semidefinite relaxation

So I'm working on the Boolean least squares problem that comes up a lot in circuit design. In its raw form, it looks like this, $$\phi = \min \operatorname{trace}(A^TAX) - 2b^TAx + b^Tb$$ s.t. $$X = ...
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Simplify Product of Sums

Similar question to: Boolean Algebra - Product of Sums I was given a truth table and asked to give the sums-of-products and the product-of-sums expressions. I reduced the sums-of-products ...
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Proving Boolean Functions

Please check my answer to this question and give me feedback . Question: Either exhibit 333 different boolean functions on the three variables p,q,r, or prove that there aren't 333 different such ...
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27 views

Boolean Functions with p,q,r

Please give me feedback for my answer to this question. Question: (1) Are the boolean functions $(p \land \neg q) \lor ( \neg r \land q)$ and $(p \lor \neg q) \land (r \lor \neg q)$ equal?. Explain ...
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Construct XNOR with only OR gates

Is it possible to construct the XNOR gate which is given as, a XNOR b = (a AND b) OR (~a AND ~b), by using only OR gates. So from the definition, the question boils down to: can you construct the AND ...
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Boolean-like algebra

Suppose one had an algebra that that follows most of the laws of Boolean algebra (associative, commutative, distributive, identity, annihilator, idempotent, double negation, De Morgan) but does not ...
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showing Boolean algebra equality

I have this exercise in my worksheet : Show that x (z ⊕ y) = xz ⊕ xy I reached this in solving it , but didn't reach the final equation x(z'y + zy') xz'y + xzy' please can someone show how
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How to prove the divisors of 15 form a Boolean algebra

This from Exercise 3.1 in "A Beginner's Guide to Discrete Mathematics" Let B be the set of all positive integer divisors of 15, that is B = {1, 3, 5, 15}. Prove that B forms a Boolean algebra with ...
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A well-defined operation on measure algebra

Let $(X,\cal{M},\mu)$ be a measure space, and for $E,F\in \cal{M}$ write $E \sim F$ iff $\mu(E \Delta F)=0$. Let $\widetilde{\cal{M}}$ be the set of equivalence classes in $\cal{M}$ for $\sim$; for ...
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Boolean function construction [duplicate]

I need some proof on this statement that not every boolean function is equal to a function constructed by only using ∨ and ∧. I need a boolean function that does not constructed using ∧ and ∨ which I ...
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99 views

Prove if Tautology, Contradicton, or Neither. Is my proof ok?

Determine whether $((p \Rightarrow q) \Rightarrow r) \Leftrightarrow (p \Rightarrow (q \Rightarrow r))$ is a tautology, a contradiction, or neither. If $p,q,r = (0,0,0)$ then $((p \Rightarrow ...
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Alternative to xor(A,B,C)

How can we make a comprehensive statement, which will correspond to the truth table of xor (A, B, C) by combining logical operators AND (&), OR (|), XOR (xor) and NOT (!)?
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436 views

Convert a boolean function into K-map

I would like to know how can I convert the following boolean function into a truth table and accordingly construct the k-map $$F = A'B'C'+B'CD'+A'BCD'+AB'C'$$ thanks in advance :)
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306 views

Boolean algebra - cube - minimal disjunctive normal form

I have a test coming up and I would like to know how to solve these kinds of problems. This is the description: ...
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227 views

Peculiar examples to the Stone Representation Theorem

The Stone Representation theorem states that every Boolean algebra is isomorphic to a field of sets. That is, a Boolean algebra whose elements are sets, and sums, products, negation are union, ...
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Is this a good enough proof?

Is this proof good enough? If not, any feedback would be appreciated. Thanks. Either exhibit $333 $ different boolean functions on the three variables $p; q; r,$ or prove that there aren’t $333$ ...
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114 views

How to prove that linear functions cannot represent binary functions

Yesterday, I thought about representing boolean algebra as linear functions: For some vector space $V$ and for some $A, B \subset V$ such that $A \ne \emptyset \,\wedge\, B \ne \emptyset ...
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If one of the hypotheses holds, then one of the conclusions holds. (looking for a proof)

Using a huge truth table, I proved the theorem below. I cannot find a more elegant proof. I tried to rewrite expressions; e.g. using the distributive laws and the laws of absorption - to no avail. Is ...
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Number of non-increasing boolean functions of $n$ booleans, up to permutations.

How many non-increasing boolean functions of $n$ boolean variables are there? I don't want to count functions that ignore some of their inputs. If two or more functions differ only by permuting their ...
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Is this *really* a categorical approach to *integration*?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
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Proving $\neg A\vee(A\wedge \neg B)= \neg A \vee \neg B$.

How do I prove using boolean algebra that $\neg A\vee(A\wedge \neg B)= \neg A \vee \neg B$? I can see it in the logic table and it is logical, but I can't prove it mathematically.
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Boolean algebra simplification question

I'm trying to simplify the follow SOP expression: $\bar{A}$$\bar{B}$$\bar{C}$ + $\bar{A}$B$\bar{C}$ + $\bar{A}$BC + AB$\bar{C}$ Using a K-map (unless I've erred) it should simplify to: ...
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51 views

Number of Positive Definite Binary Matrices

How may positive definite matrices (over finite field- $F_p$) are possible? What is the criterion in getting those?
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162 views

How do Boolean-valued functions work?

Consider this function: $$P: X\to \{true, false\}.$$ There's nothing in that expression that says when $X$ is true and when it is not true. How do these work?
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Assignment for discrete mathematics

How can I prove that not every boolean function is equal to a boolean function constructed by only using ∧ and ∨?.Need help in proving it.
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prove that there does not exists a boolean algebra containing only three element

please prove that there does not exists a Boolean algebra containing only three elements .prove it with example so that i can understand easily.i cant understand the question and i could not tried to ...
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202 views

How can I simplify this boolean equation for the multiplexer a little further?

I've obtained a formula through cannonical representation, which is: $$A\cdot \overline{B\cdot S}+A\cdot B\cdot \overline{S}+\overline{A}\cdot B\cdot S+A\cdot B \cdot S$$ And I'm trying to simplify ...
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What are three possible ways to express the following Boolean function with eight or fewer literals?

F= A'BC'D + AB'CD + A'B'C' + ACD' I assumed that the question was asking for me to simplify. I placed the terms into a kmap and have gotten SOP F= A'B'C' + A'C'D + AB'C + ACD' or POS F= ...
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Finding the atoms of a Boolean Algebra

I have a homework question that asks me to find the atoms of the Boolean Algebra that contains 256 Boolean functions "such as F1(x,y,z) = x + y +z, F2(x,y,z) = x + xz, F3(x,y,z) = xyz+ xyz and so on". ...
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71 views

Boolean functions built from $\wedge$ and $\vee$ [duplicate]

Prove that not every Boolean function is equal to a Boolean function constructed by only $\wedge$ and $\vee$. Please can you help me giving some hint.
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Logic subject-reductio ad absurdum

Can you solve this using method reductio ad absurdum? 1)A ↔ (¬ B v C) ¬ A ¬ B 2)¬(R∧ (S v T)) 3)R∧¬ T S ¬R∧ S
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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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Discrete Mathematics (boolean)

Either exhibit 333 different boolean functions on the three variables p; q; r, or prove that there aren’t 333 different such functions $p$ $q$ $r$ $0 0 0$ $001$ $010$ ...
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Checking Boolean Algebra work - Simplification

I am currently working on an assignment for a CE class I am taking, and I wanted to know if I have been simplifying these equations correctly. I'm supposed to reduce them to a sum of products. 1) ...
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359 views

Conjunctive Normal Form vs Product of Sums

I am confused as to what the difference between Conjunctive Normal Form and Product of Sums is. Can someone explain what is different about them? It seems like they both only use groups of OR ...
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Whats wrong with this reasoning…

Suppose I have two non-distinguishable balls (for example two white ones) and I color them with red and green, then a combinatorial reasoning could go like this. Suppose I enumerate the balls, ball ...
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Not every boolean function is constructed from $\wedge$ (and) and $\vee$ (or)

Prove that not every boolean function is equal to a boolean function constructed by only using $\wedge$ and $\vee$. Here is my solution, can I ask for a feed back on my solution please? $p∧q$ ...
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Boolean algebra simplification hw

I'm given the equation $F=(a+b+c)(a'+b')(a+b'+c)$ and it's supposed to simplify into a sum of two product terms, each with two literals. I know the answer is $ab'+a'c$, but I'm unsure how to get ...
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How to construct the truth table for a combinational circuit

I am trying to construct the truth table for a combinational circuit with the following conditions : A) Room with 4 doors , 1 light, a switch near each door that controls the light (4 in total) B) ...
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Simplifying Boolean Algebra Expression with 3 variables

Can someone help me simplify this in Boolean algebra? It should be one step at a time so I can understand it. The expression is: $(x+y+z)(x+z)(x'+y+z)$ I tried doing this: (it's probably wrong, ...
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A boolean algebra is complete if its stone space is extremally disconnected

I have the following proof, but I don't understand one of the steps: Theorem 4.4. A Boolean algebra is complete iff its Stone space is exlremally disconnected. Proof. Identify the given ...
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What does quotienting by a congruence mean?

I have come across quotient algebras in my different mathematics courses. I know of quotienting with normal groups, quotienting with ideals etc. While studying Boolean Algebra I encounter quotienting ...
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Discrete Mathmematics

Are the boolean functions $(p\wedge \neg q)\vee (\neg r\wedge q)$ and $(p\vee \neg q)\wedge (r \vee \neg q)$ equal? Explain your answer. Here my solution, Please give me a feed back on this solution, ...
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44 views

Homomorphism between a ring which is a boolean algebra and one which is not.

I remember reading in a textbook that there can exist a homomorphism between a ring which is a boolean algebra and one which is not. Can anyone give me some example of this.
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Sigma Algebra: Etymology

Why do we talk of sigma algebras in measure theory. As far as I know sigma is related to the countability. But what does it stand for?
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Boolean Algebra, removal of redundant term

How do I simplify this boolean expression A¬B + A¬C + BC¬D + A¬D to A¬B + A¬C + BC¬D with boolean algebra? The A¬D is redundant, I can see why it is when I examine the truth table for this ...
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Boolean Functions-Algebraic rules for Boolean functions-Associative Rule

Is the function $(p \wedge q) \vee r$ equal to the function $p \wedge (q \vee r)$? Let $a(p,q,r)=(p \wedge q) \vee r$ $b(p,q,r)= p \wedge (q \vee r)$ By associate law $a=b$, but using $a(0,0,1)=1$ and ...
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Expression conversion using de Morgan's laws

I'm sorry strongly, because it's a very dummy question... I have an example in the algebra of logic. I need to convert an expression using the rules of de Morgan - replace by the conjunction of ...
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Boolean Queries in First Order Logic

I understand first order logic and how its constructed but I have some trouble understanding how the following statement and its FO query are formed. This is from a book. ...