# Questions tagged [boolean-ring]

Use this tag for questions related to Boolean rings such as the ring of integers modulo $2$ $\mathbb Z/2\mathbb Z$.

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### Proving from De Morgan's laws

I was proving: $J.K'.L+J.K.L'+JKL=J.(L+K)$ I have done the following: From LHS: $J.K'.L+J.K.L'+JKL$ $=J.(K'.L+K.L'+K.L)$ $=J.(K'.L+K.(L'+L))$ Since $L'+L=1:$ $=J.(K'L+K)$ [' is complement] Now I'm ...
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### A question concerning supremum in a Boolean ring

In a Boolean $\sigma$-ring, is it true that for every sequence {$x_n$} and y, we have $supx_n\cdot y = sup(x_n\cdot y)$ ? Background: I was reading 'Measure Theory' written by Paul R.Halmos (in the ...
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### Number of truth outputs of a Boolean function

Disjunctive Normal Form (DNF) A Boolean function $f$ of $n$ variables is said to be in a DNF if it is a disjunction of conjunction in $n$ variables. That is, each conjunction includes all the ...
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### Number of terms in a DNF equals the number of truth outputs

In a Boolean function $f$, it seems that the number of terms in a Disjunctive Normal Form (DNF) is equal to the number of truth outputs of $f$. For example, If $f=A+B$, then clearly, $f$ has $2^2-1=3$ ...
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### Given a boolean matrix $M$ what are the matrices formed by replacing $1s$ in $M$ with $0s$ called?

Given two boolean matrices $A$ and $B$ over some common dimensions one can form an order via $A\leq B\iff \forall i,j(A_{i,j}\leq B_{i,j})$ under this order, what would the matrix $A$ be called in ...
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### Why is the boolean "OR" operator denoted as "+"?

I learnt boolean algebra as part of a computer hardware course where the focus was very much on using it as a foundation for creating digital logic blocks out of gates, so there was very early on ...
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### Equivalent properties of a proper ideal of a generalized boolean algebra

I do not understand the item c) of the following question, the exercise 9 from section 1.2 from the book "Lattice-ordered Rings and Modules" from Stuart A. Steinberg: A generalized boolean algebra is ...
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### Proving associativity of a certain binary aperation in any complemented distributive lattice

If, in a Boolean lattice $(X,\vee,\wedge,0,1,')$ (i.e. a complemented distributive lattice), we define $x+y=(x'\wedge y)\vee(x\wedge y')$, is there an elegant way to prove that $(x+y)+z=x+(y+z)$ ...
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### Rings in which each element is a sum of $n$ commuting idempotents

Let $n$ be a nonnegative integer. Let $R$ be a nonunital ring such that every element of $R$ is a sum of $n$ pairwise commuting idempotents. (As usual, the class of nonunital rings includes the class ...
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This is a short question: Assume the values: $\mathbf{True}$, $\mathbf{False}$ and the logic symbols: $\land,\lor$ Is $\mathbf{True}$, $\mathbf{False}$ under $\land,\lor \,$ isomorphic to $\... • 2,011 7 votes 2 answers 506 views ### An$m$-ary function that represents all$n$-ary functions Motivation It is well-known that any binary operator$*$on the boolean ring$\{0,1\}$can be represented using only one of the$\operatorname{NAND}$and$\operatorname{NOR}$operators. For example,... 2 votes 1 answer 200 views ### GRE 9768 #60 Boolean non-commutative rings: Prove$(-s)^2=s^2$without commutativity. GRE 9768 #60 Ian Coley's approach is to prove$(I)$and$(I) \implies (II) \implies (III)$In proving$(I)$, how do we prove $$(-s)^2=s^2$$ without commutativity (but with$s=s^2$, if need ... • 12.7k 2 votes 1 answer 227 views ### GRE 9768 #60 1. Does$(s+t)^2=s^2+t^2$imply$s+s=0$? 2. Idempotent matrices do not form a ring? GRE 9768 #60 on what appears to be Boolean rings: Ian Coley's approach is to prove$(I)$and$(I) \implies (II) \implies (III)$I think$(II) \implies (I)$. My attempt: $$(s+t)^2=s^2+t^2 \... • 12.7k -1 votes 1 answer 146 views ### Set Ring and Algebraic Ring Let \mathscr{R} be a ring of sets and define set operations \odot=\text{multiplication} and \oplus=\text{addition} by$$E\odot F=E\cap F$$and$$E\oplus F=E\triangle F$$then$\mathscr{R}$is ... • 4,337 1 vote 2 answers 533 views ### Boolean algebra/ring: Compute product of sum of combinations of variables Let's say I have a set of variables$a_1, ... ,a_n \in \{0, 1\}$and a boolean algebra where multiplication is$\land$and sum is$\oplus$(exclusive or). This is like in bit operations in computers. ... • 2,188 0 votes 2 answers 72 views ### On Boolean algebras and Atoms I know plenety of questions have been made around this topics, but i've got an specific one. If$\mathfrak A$Is an infinite boolean algebra, and$C$is a finite subset of atoms in$\mathfrak A$, then ... 0 votes 1 answer 205 views ### Are these boolean rings integral domains? Let X be a set and let P(X) denote the Boolean ring whose elements are the subsets of X, with addition being symmetric diﬀerence and multiplication being intersection. Is P({1}) an integral domain? ... • 113 0 votes 1 answer 1k views ### Show that a Boolean ring is a commutative ring. [closed] I have a question about (https://math.stackexchange.com/q/10279)'s proof to this. I also asked as a comment but I am unsure whether it will be replied to since the post was made 8 years ago? My ... 1 vote 1 answer 166 views ### How to prove convergence of a sequence of binary numbers I have a boolean expression with 4 inputs and 1 output, that when iterated onto itself(output->input(s)), the function converges to 1. How do I go about proving the convergence of a sequence of binary ... 2 votes 1 answer 217 views ### A non-Boolean ring without unity with this property I'm exploring rings that have the following property:$\rule{10cm}{0.4pt}$For all$x \in R$, and for all$n \in \mathbb{N}$, there exists$y \in R$such that$\sum_{i=1}^{n} x = y \cdot x$.$\rule{...
Let's say that f(x) is a function with its domain called $A$ with its codomain labeled $B$. The indicator function $I_{\{p\}}(x)$ has a value of the multiplicative identity, $1$, when $x=p$, and $0$ ...
Since, $a+a'=1$ and $a.a'=0$ (Boolean Algebra complement laws), where $a$ is a variable and $a'$ is its complement. Can you please explain what $a'.a'=$ would be? Also, please prove your result as ...