Articles of boolean algebra

A Distributivity Law in Complete Boolean Algebras

Let $B$ be a complete Boolean algebra. Define 3 subsets of B as follows: $B_I:= \{ u_{0,i} \mid i \in I \}$ $B_J := \{ u_{1,j} \mid j \in J \}$ $B_{I \times J} := \{ u_{0,i} \cdot u_{1,j} | (i,j) \in I \times J\}$ where $I,J$ are just some indexing sets. I am trying […]

Determining the result of Boolean shape operations on closed Bézier shapes

Given two closed shapes made up of Bézier curves (and/or straight lines), I’m looking for an efficient way of calculating the resulting shape of the following Boolean operations: union difference intersection slice (imagine each of the shapes in the Venn diagram as its own shape; this operation is optional and can be expressed as a […]

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 pair $(\alpha;\beta)$ of functions $\alpha:\mathfrak{A}\rightarrow\mathfrak{B}$, $\beta:\mathfrak{B}\rightarrow\mathfrak{A}$ such that (for every $X\in\mathfrak{A}$, $Y\in\mathfrak{B}$) $$Y\sqcap^{\mathfrak{B}}\alpha(X)\ne\bot^{\mathfrak{B}} \Leftrightarrow X\sqcap^{\mathfrak{A}}\beta(Y)\ne\bot^{\mathfrak{A}}.$$ (Boolean funcoids are a special case of pointfree funcoids as defined in […]

Can I construct a complete (as a Boolean algebra) saturated elementary extension of a given Boolean algbera?

I have been interested in the following problem: Let $B$ be an arbitrary Boolean algebra and $\kappa$ be an arbitrary cardinal. Can one construct a $\kappa$-saturated $B^* \succ B$ that is complete, i.e., all joins and meets exist in $B^*$? What are relevant sources for this problem? I bet this problem appears in very old […]

Can I construct a complete (as a Boolean algebra) $\aleph_0$ saturated elementary extension of a given Boolean algbera?

This is a follow-up to my previous question: Let $B$ be an arbitrary Boolean algebra. Can one construct a $\aleph_0$-saturated $B^* \succ B$ that is complete, i.e., all joins and meets exist in $B^*$? For my previous question, which involved uncountable cardinal numbers instead of $\aleph_0$, the answer is no. What happens we only need […]

Non-isomorphic countable Boolean algebras

I’m trying to solve the next exercise: Construct a sequence $\mathcal{B}_0,\mathcal{B}_1, \ldots$ of countable Boolean algebras such that for all $m \neq n$ then $\mathcal{B}_m \ncong \mathcal{B}_n$. I know that two countable atomless Boolean algebras are isomorphic, so I guess it has something to do with the number of atoms?! But what are examples of […]

How to Solve Boolean Matrix System?

I have a Boolean Matrix System (BMS) as described below $$Ax=c$$ where $A$ is a $n\times n$ Boolean matrix (i.e., all entries are either 0 or 1), $c$ and $x$ are two $n$-dimensional Boolean column vector. The product of two Boolean matrices $A_{m\times k}$ and $B_{k\times n}$ is $C_{m\times n}$, defined by $$c_{ij}=\bigvee_{h=1}^{k}(a_{ih}\wedge b_{hj})$$ Assume […]

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: 01) a -> separable 02) b -> separable 03) not a -> separable 04) not b -> separable 05) a and b -> separable 06) a or […]

DNF form ( $ \neg ((p \wedge q) \equiv (r \vee s)) $ )

I need your help bringing $ \neg ((p \wedge q) \equiv (r \vee s)) $ into DNF form (Disjunctive normal form). I have tried many times but appear to get slightly different answers each time. Wolfram alpha gives an answer but unfortunately no steps on how to reach this answer. I have not been able […]

If $\phi$ is a tautology then dual $\phi$ is a contradiction.

If $\phi$ is a statement form, Prove that: $\vDash \phi$ iff $\phi^{d}$ is a contradiction. where $\phi^{d}$ is the dual of $\phi .$ $\phi^{d}$ is the dual of $\phi $ is defined by: (i) $A^{d} = A$ for any statement letter A. (ii)$(\lnot \phi)^{d} = \lnot (\phi^{d})$ (iii)$(\phi \land \psi )^{d} = \phi^{d}\lor \psi^{d}.$ and […]