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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Where can I learn more about these two functions obtained from IFF and XOR?

Given a set $X$, write $\mathrm{heaps}(X)$ for the set of all finite heaps (or 'multisets', if you prefer) on $X$. Under this definition, it is well-known that if a binary operation $*$ on a set $X$ ...
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Can these nested if-then-else be turned into a boolean formula?

I have this logic statement: (A and x) or (B and y) or (not (A and B) and z) The problem is that accessing A and B are rather expensive. Therefore I'd like ...
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Boolean Algebra Manipulation/Simplification

I have come across a couple questions while doing my digital logic work. 1) Is it possible to simplify these, while keeping each a product of sums? (I'm leaning towards no--the only way I could see ...
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Boolean Function

A Boolean expression is given: (A B)’ + B C’ +A’ C = F. Construct the logical circuit and draw the timing diagram of the output F. I am not sure where to start.
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Given $x \wedge y=\mathbf{F}$, how to simplify $x \wedge \lnot y$?

Given that the boolean expression $x \wedge y=\mathbf{F}$, how to simplify $x \wedge \lnot y$? Is the above question equivalent to the following question? Find z so that $\lnot(x \wedge ...
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Can we convert this statement about sets into a statement of propositional logic?

A question was just asked here about proving $$A⊆(B∪C)⟺A\setminus C⊆B.$$ We can prove this statement directly, using the concepts of first-order logic. "Suppose $x \in A \setminus C$ and that ...
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Algebraization of attribute-value logic

Jürgen Wedekind ("Classical logics for attribute-value languages", can be googled up) has defined an attribute-value logic as a fragment of predicate logic. There are no predicates except for ...
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Boolean Algebra-Simplification Assistance Needed

I have to show that (!(P.Q) + R)(!Q + P.!R) => !Q by simplifying it using De Morgan's Laws. Here is what I did but I'm not sure it's right. (!(P.Q) + R)(!Q + P.!R) => !Q (!P + !Q + R)(!Q + P.!R) ...
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Simplifying a short Boolean expression

\begin{align*} A’B + A’B’C + ABC’C’ + AB’ + AB’C’ &= A’B + A’B’C + ABC’ + AB’ + AB’C’ \\ &= A’(B +B’C) + ABC’ + AB’(C’+1) \\ &= ??? \end{align*} I'm stuck after this. Please help me!!
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Non-Boolean group with every element of order two

Let $G$ be a group (not necessarily finite) such every element of $G$ has order 2. Every such group is abelian [1]. Clearly, every Boolean algebra $B$ is a group of this type, when equipped with the ...
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The Rotate and Shift operations in a Finite Field

Do the Rotate and Shift operations in $GF_2$ have simple expressions in a finite field? The Rotate operation $ROT[x,n]$ left rotates by n-bits. So $ROT[(0,1,1,1),2]=(1,1,0,1)$. The Shift operation ...
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Boolean bit OR operation on a Finite Field

How can I express $x \vee y$ in $GF_2$? I know that XOR is $GF_2[x]+GF_2[y]$ and AND is $GF_2[x]*GF_2[y]$ for instance. But I cannot figure out bitwise disjunction. This may be because OR does not ...
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Boolean Algebra simplification: $X=((AB)'C(A'+(B+C)'))'$

I've had two statements I need to simplify, and I'm not sure about my work: $X=((AB)'C(A'+(B+C)'))'.\quad $ With this one, do you apply DeMorgan's theorem to the interiors of the brackets and ...
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131 views

Is an algebraic formula to test real numbers equality?

Is there a formula to test numbers equality ? Let $x$ and $y$ real numbers. If $x=y$ the formula will results $1$. Else the formula will results $0$. I'm not searching for an algorithmic solution ...
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578 views

Boolean Algebra Simplification - In sum of products form

How would you simplify this expression? I've been struggling with it for a while, but seem not to be getting anywhere near the right answer. Y = (A' + BD + C'D)' (B'CD')
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88 views

How are boolean expressions converted to NOR expressions?

What kind of rules help to convert an expression into a 3 input NOR expression? Do all variables have to be of the form (a+b+c)' + (d+e+f)'? What happens if there is an expression that is just (a')' ...
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65 views

Deriving truth table from English description

I'm trying to check if my truth table is correct since it largely depends on further parts of a larger problem. Here is the English description: The controller will turn on the headlights under the ...
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102 views

What is a (-1)-morphism?

So, I read the John Baez essay "Lectures on n-categories and cohomology" and I understand the notion of a (-1)-category" and a (-2)-category" and how to derive them. However, I'm not totally clear on ...
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Cardinality of the set of ultrafilters on an infinite Boolean algebra

Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?
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If $G$ is a generic ultrafilter, why $(\exists a\in A)(a\in G)\leftrightarrow \Sigma A\in G$?

Let $B$ be a complete Boolean Algebra. Let $G$ be a generic ultrafilter of $B$, that is, such that for any dense $D\subset B$ we have $D\cap G\neq \emptyset$. Why for all $A\subseteq B,$ $\Sigma A\in ...
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350 views

Number of canonical expressions

There is a question: What is the number of canonical expressions that can be developed over a 3-valued boolean algebra? I was trying to solve this. Canonical expression is the combination of ...
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Fourier transformations in Simon's quantum paper

I am reading this paper by Simon. This is one of the earliest quantum algorithm papers. In the paper he presents a routine starting at the end of page six. The first step runs a Fourier transformation ...
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131 views

boolean algebra: DeMorgan's law confusion

the following function should be put into table values: $$y = \overline{(a*b*d+c)}$$ So the first thing i am doing is using DeMorgan to get rid of the "whole-term-negation": $$y = (\tilde a + \tilde b ...
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How to write boolean expression as linear equations 2

I just posted How to write boolean expressions as linear equations and asked about a simple example. Here's what we know so far: Suppose a,b,c,d,e ∈ {0,1}. if the boolean expression is: a ≠ b, I ...
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boolean algebra: simplify $ a* b *d + \tilde a *\tilde c*d + b* \tilde c* d$

Simplify the following function(algebraically): $$y = a*b*d + \tilde a *\tilde c*d + b *\tilde c *d$$ the solution is: $$a*b*d + \tilde a * \tilde c * d$$ which i checked via karnaugh and also ...
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50 views

Larger circuit design for same boolean function?

I've designed this circuit with 4 logic gates, and did Karnaugh map's simplification and Quine McCluskey method. However I found out that actually my circuit design is already optimized and I can't ...
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35 views

Distinct Karnaugh Maps grouping?

I got a table truth with some minterns which I mapped to a Karnaugh Map, then I can see an obvious choice for grouping. But I'm wondering wether in this case is possible to do any other different to ...
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61 views

Help at solving boolean function.

I`m having some difficulties solving a boolean expression (I am converting it to CNF form). The expression is: $$F = (Q_1 \to P1 \land \lnot P_2) \lor Q_1 \land P_2 \lor P_1$$ So i do not know, ...
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138 views

How to write boolean expressions as linear equations

I want to convert a set of boolean expressions to linear equations. In some cases, this is easy. For example, suppose $a, b, c$ $\in$ {0,1}. Then if the boolean expression is: $a$ $\ne$ b, I could use ...
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Is there a way to prove a boolean operator isn't universal?

In boolean algebra, I could prove an operator is universal by implementing a NAND or NOR gate with it. But is there a way to prove a boolean operator isn't universal? I would like to know a general ...
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151 views

Proving $(xyz)' = x'+y'+z'$

I'm trying to prove that $(xyz)' = x'+y'+z'$ using theorems/axioms. I tried this but I just want to make sure if its the correct route or if I've done something "illegal"/wrong. ...
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108 views

Binomial expansion through combinations.

If you have $(a+b)(c+d)(e+f)$ how can you expand this? Someone was mentioning that you get different combinations so that you get $adf+ade+acf+ace+bdf+bde+bce+bcf$? Is that the full expansion? As an ...
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190 views

Reduce to sum of products

I'm given the equation $F = (x+w)z' + x(y+z) + xz$ The inverse I got is $F' = [(x'+z)*(w'+z)]*[(x'+y')*(x'+z')]*[x'+z']$ To start I would expand $F'= [x'w' + x'z + zw'+zz] * [x'x' + x'z' + y'x' + ...
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Understanding Sum of product and complete sum of product

I have a pair of problems, the first two of my homework, and I'm already unclear on how finding SOP and CSOP for them work. The first: E=xy(1+z)y' It seems like this just reduces to 0, since 1+z ...
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Infix to Postfix

4 + x / b - a + 5 AND x AND y OR p OR q What is the tree and the postfix of the expression above? I find it tricky because I am not sure if AND has higher precedence than the arithmetic operators, ...
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Proof of $(P\Leftrightarrow Q)\Leftrightarrow((P\Rightarrow Q)\wedge (Q\Rightarrow P))$

Hi I've been working through Applied Mathematics for Database Professionals and I'm stuck trying to proof this equivalence: $$(P\Leftrightarrow Q)\Leftrightarrow((P\Rightarrow Q)\wedge ...
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204 views

Single Complement Variable + 1

Is a complement + 1 = 1? For example A' + 1 = 0; I was thinking it was (I'm new to boolean algebra) since A' = 0, and 0 + 1 in boolean algebra is just 1. Of course, A can be anything, but assuming ...
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80 views

Multiply the number $(1001)_{2}$ by 3 digit number

I want to multiply the number $(9)_{10} \rightarrow (1001)_{2}$ by a 3 digit binary number. 1) How I can extract the boolean equations? 2) Make a circuit of it. so what I did is just see what ...
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Can a Karnaugh map be solved in more than one way?

So I understand for doing a simplification by a K Map I should group my 1's (or even 0's) in $2^n$ elements, always trying to grup as much elements as possible. And I can group even adjacent elements ...
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Is it possible to derive all the other boolean functions by taking other primitives different of $NAND$?

I was reading the TECS book (The Elements of Computing Systems), in the book, we start to build the other logical gates with a single primitive logical gate, the $NAND$ gate. With it, we could easily ...
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106 views

Can't simplify this boolean algebra equation

So I've got an expression I have been trying to simplify and have the answer but I can't figure out how to get to it... can anyone help me out? Equation: $(A\wedge \lnot B \wedge \lnot C \wedge ...
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400 views

How many presentable boolean functions with n attributes are linear separable?

The aim is to find a formula for the question. For n=2 i get (2^(2^n)=16 possible functions. This is the solution for a boolean function with 2 attributes: ...
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Logic Circuit Question

1) Write the boolean expression after every GATE 2) Write the boolean expression of GATE 3 3) Try to simplify the boolean expression of GATE3 I need to know if what I did its right + your advice ...
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Boolean simplification, 5 variables

I'm currently learning for my maths exam, and in the part about boolean algebra I came across an exercise that I can't seem to solve. I probably only need the first few steps to get started. $$ (xyz ...
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Stone's Representation Theorem and The Compactness Theorem

If you're working on $\mathsf {ZF}$ and you assume the compactness theorem for propositional logic, then you have the prime ideal theorem, and thus you can show that the dual of the category of ...
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Simplify Boolean Algebra

How do I simplify the following expression with Boolean Algebra? Please show what you used to simplify so I can understand. $$ABC + AB'C' + ABC' + A'B'C'$$
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The input represent a 4-bit unsigned binary number, the output W, is 1 if the number is multiple of 2 or 3 but not both.

I completely understand how to make a truth table and the entire concept of boolean algebra. However, I am confused how to make the truth table for the above information. Because the input is a 4-bit ...
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maximal antichain

I don't understand the definition of Jech (set theory) for "maximal antichain". Let $B$ a boolean algebra and $A$ a subalgebra of $B$. $W\subseteq A^+$ is a maximal antichain if $\sum W=1$ and $W$ ...
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Why is $ab + bc + c\bar{a} = ab + c\bar{a}$ true in binary?

I was simplifying the equation of a logic gates problem and I realized that ab + bc + cā and ab + cā followed the same truth ...
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Which law of logical equivalence says $P\Leftrightarrow Q ≡ (P\lor Q) \Rightarrow(P\land Q)$

I'm going through the exercises in the book Discrete Mathematics with Applications. I'm asked to show that two circuits are equivalent by converting them to boolean expressions and using the laws in ...