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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Why does the number of 1s in a prime implicant set in a Karnaugh Map need to be a power of 2?

Pretty much the title. We were learning about Karnaugh maps in class today and they didn't really mention why it has to be a power of 2. A quick google search basically confirmed that it needs to be a ...
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45 views

simple logic to use only one operator to construct a function [closed]

So I was doing some of the problems on the book for discrete mathematics and I encountered this problem: We define a new operator ⊙ as follows: x ...y.... x⊙y F... F... T F... T... F T... F... F ...
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Looking for a particular algebraic mapping from one Boolean matrix to another

Consider the following Boolean matrix: \begin{align} X&=\begin{bmatrix} 1&1&1&1&1&1&1&1\\ 1&1&1&1&0&0&0&0\\ ...
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finding shortest equivalent expression

I am trying to find the shortest equivalent expression of the following: ((C → D) $\wedge$ (D → C)) $↔$ (C $\wedge$ D ∨ ¬C $\wedge$ ¬D) I have "simplified" the expression into the following: ...
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functional completeness of $\{\to\}$ [duplicate]

Given that the set {∨, $\wedge$ , ¬} is functionally complete, how would I prove whether the set $\{\to\}$ is functionally complete? expressing $→$ in terms of $∨$: $¬A∨B$ expressing $→$ in terms ...
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proof of functional completeness of logical operators

If I know that the set of operators {∨, & , ¬} is functionally complete, how do I go about proving/disproving the functional completeness of the following set of operators? a) $\{\vee,\neg\}$ b) ...
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119 views

Prove that the set {→, ¬} is functionally complete

I am not sure how to do this question. I have looked at some of the other similar questions but to no avail I know that for a set of operators to be functionally complete, the set can be used to ...
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53 views

Essential Prime Implicants confusion

I'm trying to understand what an essential prime implicant is. This picture says that there are 2 essential prime implicants but why can't we say that there's none? Can't we cover those ones ...
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how to determine if formula satisfies without creating a truth table

$(p \wedge q \wedge r) \wedge (\neg p \vee r)$ So far, what I have got is that $(p \wedge q \wedge r)$ satisfies because if $p$, $q$ and $r = 1$ then $(p \wedge q \wedge r)$ also $= 1$. For $(\neg p ...
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84 views

Simplifying a Boolean expression for two-level NAND gate circuits

The expression is: F = (X' + Y' + Z')(Y' + A') I have no clear idea on how to go about simplifying this with Boolean algebra. After it's simplified, I'll need to implement it only using NAND gates. ...
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56 views

Minimizing using a Karnaugh map when given as subscripts F4,2655

I have to minimize the expression using minterms and a Karnaugh map: $F_{4,2655}$ How might I get this expression I am given into a form much like a typical boolean algebra minification question? I ...
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Embed boolean lattice into complete atomic boolean lattice

Trying to answer this question, I attempt to apply the solution of this question. To use this way, I may need to embed a boolean lattice into a complete atomic boolean lattice. Please help to ...
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1answer
54 views

Is canonical SOP/POS form for a boolean expression unique?

I was trying to find equivalence between two boolean expressions and thought if I convert both of them to canonical sum of product or product of sum form they should match. But I am not sure if these ...
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1answer
24 views

boolean algebra simplification solving

Can anyone help me out on this boolean algebra simplification...im not sure with my answer. X’YZ + XY’Z’ + X’Y’Z’ + XY’Z + XYZ my answer is x'yz+y'z'+xz but badly not sure of it! can you check thnks ...
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31 views

Boolean Simplification questions

I'm having some trouble getting a handle with this course. We are starting Boolean algebra and my professor wants us simplify the following: Im sorry for the ignorance but I can't find a good ...
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68 views

Boolean algebra: How does the imply operator work?

I am conflicted because our professor said that "(not) A implies B only and only if A and B = 0" which doesn't match with what I found on Wikipedia or other books on Boolean algebra: "A implies B = ...
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2answers
57 views

Name for a Boolean ring without a unit element

Is there a standard name for a Boolean ring without a unit? I read that historically ring and Boolean ring used to refer to possibly non-unital objects: The old terminology was to use "Boolean ...
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21 views

Simplify Boolean Expression

so I have this expression and I have to simplify it to minimum SoPs $(x+(y'(z+w)')')'$ so my final answer is $x'y'z'w'$ but I think there is something wrong or trick can some one help me or tell ...
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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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33 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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1answer
204 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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204 views

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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27 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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26 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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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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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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129 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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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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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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62 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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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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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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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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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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58 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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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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202 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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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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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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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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146 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 ...