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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Is D646 a Boolean Algebra?

I read here: http://mathoverflow.net/questions/193924/how-to-recognize-if-a-lattice-is-distributive?newreg=1439abdc43e24ebcb32afa0532b74ecb that N5 and M3 lattices are not distributive. So I concluded ...
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29 views

How are these two boolean expressions same?

How does AB(1+C'D) simplify into AB in boolean algebra? I cannot compare their truth tables since literal number of these two ...
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130 views

Duality Principle vs. DeMorgan Law

What is the difference between the two? Duality Principle states that any theorem in switching algebra remains true if 0 and 1 are swapped and + and . are swapped throughout. DeMorgan's Law says ...
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Finding minterms from a boolean expression

I have a question regarding the process of finding minterms. Problem: Find the minterms of the following expression by first plotting each expression on a K-map: ...
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23 views

Construct a set of 4 elements and an operation * that is closed, with universal identity, no universal inverse. Can it be commutative?

I am very confused on this problem I have for math. Constructing my own table for this as well as determining identities and inverses leaves me clueless. Any help would be greatly appreciated!
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26 views

Simplify Boolean Expession

Can anyone verify this. If I can wrong can you point me in the correction direction: $$AB'C'+A'B'C+A'BC'+AB'C = B'(AC'+A'C+AC)+A'BC' \rightarrow B'(AC'+C)+A'BC' \rightarrow B'(C+A)+A'BC'\rightarrow ...
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Poset is complete iff it is cocomplete

In Awodey's Category Theory, page 130, he says: A poset is (co)-complete if it is so as a category, thus if it has all set-indexed meets (resp. joins). For posets, completeness and cocompleteness ...
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Convert a'+b(a+b')(b+c') to sum of products and product of sums using boolean algebra.

Sum of products F = a'+b(a+b')(b+c') = a'+ ab + bb' (b+c') = a'+ab So that's about as far as I've gotten. I'm trying to find a step by step guide on how to ...
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22 views

CNF form in Boolean algebra

I have problems with CNF form of formula in boolean logic. I need to get it using only laws of boolean algebra. The formula is: $$(!a \land !b \land !c) \lor (!a \land !b \land d) \lor (b \land c ...
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32 views

Write all axioms and properties for the Boolean algebra of sets P(S) (power set)

Write all axioms and properties for the Boolean algebra of sets: $S = set$ $(P(S), \cap$$, ∪ , complement; ∅, S)$ I know the axioms of Boolean algebra but I am not sure how to translate that to a ...
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30 views

Mathematics of two's complement

I am trying to understand the underlying mathematics of two's complement. Googling the topic gives me a lot of articles on how to invert the digits and add one, and why computers use this system ...
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70 views

Sigma notation with minterms

I'm trying to understand what is meant by the notation $F(x,y,z)=\sum m(0,1,2,3)$ I found this webpage but it's still unclear to me. It states the function F is defined by the truth table ...
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1answer
14 views

Boolean algebra simplifcation

((q IMPLIES p) OR ((r OR q) AND (NOT q OR p) )) AND ((NOT p AND q) IMPLIES r) how do I simplify this to (p or not q) I'm stumped I have tried applying rules of interference but I just cant get it a ...
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1answer
45 views

Logic - Propositional calculus

I don't understand how to show the following: (!Q -> P) ∧ !P -> Q I understand the answer is true as I did it with a truth table but how can I prove this using propositional logic? Thanks!
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58 views

Understanding how $\mathcal{P}(A)$ is a Boolean algebra

I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding Boolean algebra. To be specific, I'm stuck on the following practice question: ...
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52 views

Boolean Ring and prime ideals

Referring to the question: Finitely generated ideals in a Boolean ring are principal, why? How to prove: In every Boolean Ring Does there exist any prime ideal in a Boolean Ring. Only Boolean ring I ...
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149 views

A construction on boolean lattices is itself a boolean lattice?

Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) boolean lattices (with lattice operations denoted $\sqcup$ and $\sqcap$, bottom element $\bot$ and top element $\top$). I call a boolean funcoid a ...
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45 views

Find the complement of Boolean function [closed]

I have a boolean function, C + (A ⋅ B) However, I am not sure how to find its complement. I have thought about inverting it twice, but that re-simplifies to the same original, and I have searched ...
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40 views

Why does (A'+ AC) = (A'+C)?

Why does (A'+ AC) = (A'+C)? I can understand this via a truth table, but I cannot see why this works in boolean algebra. Conceptually I understand that A doesn't matter, but I can not seem to prove ...
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1answer
17 views

Does there exist a boolean lattice without atoms?

There are atomic boolean lattices and this is the same as atomistic boolean lattices. Does there exist a boolean lattice without atoms? (except of the degenerative case of one-element lattice) Or at ...
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25 views

Reducing boolean algebra

So I got this equation: (NOT A + B) x ( A + C) When I try to reduce this I get (Not A AND C) OR (A AND B) OR (B AND C) But ...
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1answer
26 views

Coming up with an expression given the truth table

I have been given a truth table. I want find a boolean expression for it. However , I am not able to come up with one. Is there a specific way to go about , in order to get it done ? Also, Can this ...
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2answers
46 views

Understanding boolean algebra and boolean axioms?

I'm currently studying discrete mathematics and am having some difficulties with understanding boolean algebra. To be specific, I'm stuck on the following question: ...
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2answers
23 views

question about the notation in boolean algebra?

Is there a difference between $\bar{A}\bar{B}$ $\overline{AB}$ Is there a difference between $\bar{A}+\bar{B}$ $\overline{A+B}$ Also, just to be sure, the equal sign is a normal equal sign in ...
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trying to simplify equation to fullest

I've got an equation down to $F=BA+BC +\bar{A}\bar{C}$ and according to Wolfram Alpha it can simplified to $(\bar{A} + \bar{C}) + B$. What's the next steps? I tried using de Morgan's law and not sure ...
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1answer
152 views

sum of products Boolean algebra simplification

I have a question that states the following: Use algebraic manipulation to show that for three input variables x1 , x2 , and x3 ∑(1,2,3,4,5,6,7) = x1 + x2 + x3 ...
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Simplifying a 4-term equation using boolean algebra

1432928 So, I have a logical expression: $(\lnot A\land B\land C\land \lnot D)\lor (\lnot A\land B\land C\land D)\lor (A\land \lnot B\land \lnot C\land D)\lor (A\land \lnot B\land C\land D)\lor ...
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Question on free Boolean algebras

Every Boolean algebra $A$ is isomorphic to a field of set. In particular, if $A$ is finite, then $A$ is isomorphic to the power set of its atoms. Now, suppose that $A$ is free Boolean algebra with 2 ...
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1answer
27 views

Nullary and unary operations defined on a uniquely complemented lattice?

A lattice is a set $L$ with two binary operations, $\lor$ "join" and $\land$ "meet". In a complemented lattice, for every element $a$ there exists an element $a^{\perp}$ such that $a \lor a^\perp=1$ ...
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36 views

Explain why this is a Boolean algebra.

There is a couple answers to this already, which have helped me get this far. But I have a few questions left. Is there a rule in Set Theory that says I have to assign the list to a "Set" before ...
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1answer
27 views

A question in boolean calculus

I've looked everywhere on the web, in my note and to fellow student without any of them able to explain to me. Here is my question to you. I can't figure out how to calculate this boolean expression ...
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1answer
98 views

Duality principle in arithmetic

How do I show that if you have a valid rule in arithmetic that involves multiplication and addition, then you cannot interchange the signs of multiplication and addition and obtain a valid rule? I ...
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Boolean algebra: why is $a\overline bc + ab = ac+ab$?

Why is $a\overline bc + ab = ab + ac$? I think it has something to do with the rule $a + \overline a = 1$, right?
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Assistance in proving a tautology using a series of logical equivalences.

I am trying to prove, using a series of logical equivalence rules, that the following formula is a tautology: $$[a∧(a→b)∧(b→c)]→c$$ Yet despite numerous successes on other tautologies and logical ...
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1answer
19 views

finite-state machine for a system

Every cycle we get a bit $x_t$. We output $1$ iff $$(x_1\ldots x_t) \bmod 5 = 2 \lor (x_1\ldots x_t) \bmod 5 = 4$$ I need to design an FSM (preferably Mealy machine but that doesn't really matter. ...
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Hadamard matice decomposition to sparce matrices

$H_2=\begin{pmatrix} 1 & 1\\1 & -1 \end{pmatrix}$ and $H_{2n}=H_2\otimes H_n$. I am looking for decomposition of $H_n$ to sparce matrices and its proof. Is there any good source to suggest ? ...
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53 views

Boolean Algebra Karnaugh Maps

I'm having trouble solving this: Simplify the expression F = W'X'Y'Z' + W'X'YZ' + WX'Y'Z' + WX'YZ' + WXYZ + W'XYZ using a Karnaugh Map. The book I have very poorly describes how to do Karnaugh Maps. ...
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1answer
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A question about a generated $\sigma$-algebra of a family set

Wikipedia's definition of Family of sets: In set theory, a collection $F$ of subsets of a given set $S$ is called a family of subsets of $S$, or a family of sets over $S$. So suppose $Ω$ is ...
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1answer
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Proving $(p\to q)\lor (r\to s) \vdash (p\to s)\lor(r\to q)$ using Fitch notation

I'm supposed to prove the validity of the following $(p\to q)\lor (r\to s) \vdash (p\to s)\lor(r\to q)$ I'm very new to natural deduction, so I still haven't got a "feel" about it. I can prove ...
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288 views

Build a 3 bit full adder using only XOR gate?

I don't know if this is the right place to ask this, but I'm trying to design the logic for a simple calculator and I was wondering how can you build/design a 3 bit full adder using only XOR (one or ...
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1answer
74 views

extending automorphisms in complete boolean algebras

Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$. Suppose $f : A \to A$ is an automorphism. Then $f$ can be extended to an automorphism of $B$. I can see this using the fact ...
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Countably closed Boolean algebra of subsets of the real plane,

The following problem was in The American Mathematical Monthly : A generalized rectangle is $E \times F$ for any subsets $E,F$ of $\Bbb R$ (the reals). If $\mathscr{B}$ is the smallest countably ...
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Simple Boolean Algebra Exercise but stuck

I have the following exercise that I can't really solve or I am not happy with the result: If Team A loses, Team B and C will lose too If the Teams A and B win, Team C will lose If Team B ...
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31 views

Notation of a boolean function

I'm studying Boolean algebra but I was confused as the notation of a Boolean function. When I write/denote a Boolean function that way, what does that mean? $$ f: \mathbb{Z}^2_2 ...
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Solving boolean equation

Assume we want to solve, with f and g boolean functions f=-g' this has the same solutions as f(-g')'+f'(-g')=0 -fg-f'g'=0 Is this statement correct or am i completely wrong?
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Boolean algebra laws

Can someone explain to me why in Boolean algebra $$ f(x,y,z,t)=z+x'y+xy'+xt'+yt' =z+x'y+xy'+xt'$$ I have no clue why u can just leave out the last term, is it due to some ...
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1answer
28 views

Boolean Algebra and negation

$$ -(A \ast -B) \ast -(-A \ast B) $$ My understanding is that the above logic is equal to $$ (-A \ast B) \ast (A \ast -B) = (-A \ast A) \ast (-B \ast B) = \mathrm{FALSE} $$ but my ...
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Generators of free Boolean algebras

Suppose $\mathfrak{A}$ is a free Boolean algebra and $G$ a countable set of free generators of $\mathfrak{A}$. What is the cardinality of $\mathfrak{A}$ if $G$ is countably infinite, but we only ...
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108 views

Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
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Boolean Algebra Problem ABCC' [closed]

Hi I just want to ask the answer of this Boolean Algebra problem.. $$ABCC' + B + A'B $$ How to simplify that one?