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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Logical limitations of Proofs by Contradiction

In general proofs by contradiction go as follows: Given an arbitrary hypothesis, $\ p \implies q$, we assume $\left(p\implies q\right) = T$, and then we show that by assuming the hypothesis to be ...
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Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$?

If I use the above statement, provided that it is right, in a question, would I have to prove it as well?
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Minimize SOP and POS algebraically?

Is it possible to simplify an SOP (sum of products) or POS (product of sums) expression algebraically? I can only do it through k-maps. Example: $a'b'c'd' + a'b'c'd + a'b'cd' + a'b'cd + ab'c'd + ...
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How to deal with an 8 variable Karnaugh map

I'm reaching back into my high school days trying to remember one of the rules about Karnaugh Maps. I have an 8 variable input, and I remember that I should try and make the selections a big as ...
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Is there a blackboard bold letter for the set of Boolean numbers? [duplicate]

Is there a symbol (e.g. $\mathbb{B}$) for the special set of Boolean numbers or values; ${0,1}$ or ${True,False}$?
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Stuck at simplifying boolean expression

I'm getting stuck at the following boolean expression. $$z + (x'y) + (xy') + (xt') + (yt')$$ In my solutions it's simplified and the $(yt')$ term is gone. How do they simplify this? I really cant ...
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Galois field of order 2 constituting a Boolean algebra

We know that the the set $\{0,1\}$ constitutes a Boolean Algebra over the usual $OR$ and $AND$ operations. However, because of the lack of an additive inverse for $1$ this does not produce a Galois ...
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A proof of maximality of an antichain in a complete Boolean algebra

Suppose $(\mathbf{B},\leqslant)$ is a complete Boolean algebra and let $|\mathbf{B}|=|\gamma|$. Let $C=\langle c_\alpha\mid \alpha<\gamma\rangle$ be a maximal descending chain without a lower ...
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Boolean Expression Simplifying explanation

Currently have worked xz' + x'y + (yz)' Down to z' + x'y + y' Is this its simplest form? METHOD: xz' + x'y + (yz)' -> De-Morgan on (yz)' xz' + x'y + y' + z' -> Commutative xz' + z' + x'y + y' -> ...
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Do DeMorgan's laws hold for pseudo-complement in Bi-Heyting Algebra?

A textbook says in Heyting Algebra, The pseudo-complement of an element $a$ is denoted as $a^{\ast}$. One of the DeMorgan's law $\left(\vee a_{i}\right)^{\ast}=\wedge a_{i}^{\ast}$ holds ...
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Is a boolean algebra closed under countable disjunction/conjunction?

I'm just curious if the properties in a sigma algebra is also satisfied in a boolean algebra. In a boolean algebra, the two operators are closed under finite operations, but can we say they are closed ...
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boolean algebra reduction question

hi im having a lot of trouble proving this boolean expression. Im getting many differing answers so I assume I must be going about it in the wrong way. To explain, I'm trying to negate the whole LHS ...
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Boolean Logic - Why doesn't the Associativity Law work in this scenario?

By the associativity law, shouldn't the statement below be true? I understand that the truth tables are different but where exactly does the associativity law apply then if this is False? $(p \land ...
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Why are Boolean Algebras called “Algebras”?

Boolean algebras aren't algebras (to the best of my understanding). So why are they called algebras? Wouldn't it make more sense to call them a "Boolean system" or a "Boology" or something else like ...
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General rules for transforming boolean equations?

Are there general or restricted rules for transforming between equivalent boolean equations? A concrete problem that I have is given the following equation: ...
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Help with Simplifying boolean algebra, not sure if i have done it correctly.

I have no idea how to do boolean algebra, First question is x'y + x(x + y') I need to first draw a circuit diagram(logic gate) and then simplify it and draw a simplified logic gate. As of now I ...
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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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Comparing entropies $H((f(X,Y), g(X,Y)))$ and $H ((f(X,Y),g(X,Z)))$

Let X,Y,Z be three independent uniform distributions on $\{0,1\}^n$; $f, g:\{0,1\}^n\times\{0,1\}^n\rightarrow\{0,1\}$ be two boolean functions. Is it true that $$H((f(X,Y), g(X,Y)))\leq H ...
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Correct definition of the co-occurrence graph of a pseudo-Boolean function

In section 4.6 of Pseudo-Boolean Optimization, Boros and Hammer have defined the co-occurrence graph of a pseudo-Boolean function as follows. If a pseudo-Boolean function $f : \mathbb{B}^n \mapsto ...
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how to construct a boolean algebra out of a set of well formed formulas?

Given a set of well formed formulas of a first order language (with equality, constants, variables, non-logical symbols, etc), is it possible to use it as some kind of base to construct a (possible ...
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What is the name of a “basis” in Boolean algebras

So, a basis in linear algebra is the smallest set which generates a particular vector space. (More formally, a subset of the vector space which is linearly independent and spans the vector space) Is ...
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Is there an $O(n^2)$ test to determine if an $n \times n$ Boolean matrix $B$ has an inverse?

D.E. Rutherford shows that if a Boolean matrix $B$ has an inverse, then $B^{-1}= B^T$, or $BB^T=B^TB=I$. I have two related questions: The only invertible Boolean matrices I can find are ...
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Simplify problem using duality or consensus theorem

I have a difficult with the problem, whenever I try to solve this problem $W'Y'+WYZ+XY'Z+W'X'+Y$ (three literal term) I confused with so I have to factor equation to simplify,and I quite confused ...
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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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What is wrong with my Boolean expression?

I've got the following expression: A*!B*(!B*C+!C*A*(D+B*!A+D*A*B+C*!D)) After translating it to Wolfram's understandable language I got this: ...
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389 views

Memory and bits. Need some help

Could someone check over my answers to verify I am correct. Say we have a memory consisting of 2048 locations, and each location contains 16 bits. ◦ A) How many bits are required for the address? ...
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132 views

Can $P(\omega)$ be superatomic?

A Boolean algebra is superatomic if its every subalgebra has an atom. I'm trying to determine whether $P(\omega)$, i.e. the power set algebra of the set of all natural numbers (finite ordinals) ...
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Is every quotient algebra of a Boolean algebra isomorphic to a subalgebra?

Is every (non-trivial) quotient of a Boolean algebra isomorphic to a subalgebra of that Boolean algebra? And conversely is every subalgebra isomorphic to a quotient algebra?
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What is this product called?

Let $X$ be a finite set and let $2^X$ be its power set. Let $Z$ be some ring (e.g. the complex numbers; it doesn't matter). Suppose $f:2^X\to Z$ and $g:2^X\to Z$ are two functions from $2^X$ to $Z$. ...
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Confirmation of an answer of a question on Boolean Algebra

Here are the solution I have worked out. Is it correct? Given $C + BC'$: $C + B' + C'$ $C + (B'+C')'$ $C + B + C$ $C (C + B + C)$. Is the answer (2)?
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$\vDash \varphi$ iff $\| \varphi \|_A =1$ for every boolean valued structure $A$

In the book Axiomatic Set Theory (Takeuti, G; Zaring, W.M. - 1973) the theorem 6.4 states that if $\varphi$ is a closed formula of a given language then it is satisfied in every boolean valued ...
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How to find standard product of sums?

I have came across an exercise in book to find the standard product of sums with the following function: F(A,B,C,D,E,F) = (A + BC'+ CD) (B' + EF) Here's my approach to solve it: Step 1: [ (A + B) ...
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Reducing a Boolean function

I have the following boolean function: f(x,y,z) = xyz + xyz' + xy'z + x'yz + xy'z' I could reduce it to the following: f(x,y,z) = xy + xy'z + x'yz + xy'z Im not sure what to do next, i know it can ...
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help on simplifying boolean algebra

I need t show the the terms on the left simplify to the ones on the right $$(X+Y).(X'+Z)= X.Z+X'.Y$$ My attempt: I went with $$XX'+XZ+YX'+YZ= 0 +XZ+YX'+YZ$$ But I'm stumped beyond ...
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Algorithms - Finding Clique of size n in a Graph

I have the following statements (NOTE: $\bar x$ means the complement of $x$): $(x_1 V \bar x_2 V x_3) ∧ ( \bar x_1 V x_2 V x_3) ∧ (x_1 V \bar x_3) ∧ (x_2 V \bar x_3 V x_4)$ I need to do the ...
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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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Boolean functions

Boolean function f(x1,x2,x3): If f(x1,x2,x3)= TRUE then f(TRUE,x2,x3)= TRUE ...
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Is this the correct way of drawing a combinatorial circuit based on the disjunctive normal form and logic table?

The logic table: $$\begin{array}{|c3:c|}\hline x & y & z & f(x,y,z) \\\hline 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & ...
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Tests for combinational logic - product of sums simplification

I am trying to understand how to simplify the Boolean cover function $(4+6)(12+14)(0+1+2+3)(11)(3)(9)(13)(10+14)(1+3+9)$ which is a stuck-at fault test cover function derived in this paper. I know ...
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Simplify a Boolean Algebra expression with don't cares

In my homework assignment, I'm asked to simplify an expression of Q'RS'T' + Q'R'S'T + RS'T with don't-cares of m3, m12, and m14. I know how I would achieve this result with a K-map, however the ...
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Solving $n$-binary-variable system of equations using only combinations of $n \over 2$ variables when $n \over 2$ is even

It seems that it's impossible to find the unique solution to an $n$-binary-variable system of XOR equations if you only use all $(n \text{ choose } {n \over 2})$ equations combining half the ...
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necessity of $f(0)=0$ and $f(1)=1$ in homomorphisms of boolean algebras

Let $A,B$ be boolean algebras and let $f \colon A \rightarrow B$. $f$ is a homomorphism of boolean algebras if $f$ is a homomorphism of the corresponding lattices and $f(0)=0$ and $f(1)=1$. Why is it ...
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Boolean function analysis on random graphs?

Random graphs have some properties that are determined in some random way such as edge probabilities in the interval $[0,1]$. Ryan O'Donnell's book "Analysis of Boolean Functions" (2014) has analysis ...
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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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number of permutation in a boolean expression containing only ANDs and ORs

I need to find the number of permutations of some expression which contains only conjunctions and disjunctions e.g.: $$ e = x_1x_2 \vee x_3x_4 $$ where $x_1x_2$ and $x_3x_4$ are boolean summands, ...
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Pseudo-Boolean functions restricted to integers

The Pseudo-Boolean functions are of the following form. $$ f : \mathbb{B}^n \to \mathbb{R} $$ I would like to know if there is a special sub-category of $$ f : \mathbb{B}^n \to \mathbb{Z} $$ with ...
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What will be the answer to this K-Map?

I have a K-Map and I need to figure out which expression isn't equivalent to the provided K-Map. ...
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59 views

Are all algebras groups?

It seemed to me that boolean algebra is a group because it is closed (You can't use boolean algebra and get a result that is outside the group) under a logical primitive(?) and order of operands and ...
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Boolean Algebra laws of deduction question

I have a question in which I'm a little stuck at answering, could anyone help? ...
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Want to check if my Boolean Algebra simplification is correct

$(A+B)(B+\bar B)(\bar B+C)$ Distributive LAW $(AB+A \bar B+B B+B \bar B)(\bar B+C)$ Distributive LAW $(A B \bar B+A B C+A \bar B \bar B+A \bar B C+B B \bar B+B B C+B \bar B \bar B+B \bar B C)$ ...