# Questions tagged [boolean-algebra]

Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Use this tag for questions about Boolean algebras as structures, or about functions defined from/to Boolean algebras. For Boolean logic use the tag propositional-calculus.

2,902 questions
Filter by
Sorted by
Tagged with
12 views

### please solve this $F = xyz + xy' + x'y'z' + xyz'$ [closed]

please solve this $F = xyz + xy' + x'y'z' + xyz'$
1 vote
12 views

### Isomorphism between Boolean algebras $D$ and $B * (D : B)$ (Exercise 16.4 of Jech)

Let $B,D$ be two complete Boolean algebras in $V$ such that $B$ embeds into $D$ as a complete Boolean subalgebra. Let $V^B$ be the Boolean-valued model (w.r.t. $B$). Let $\dot{G}$ be the canonical ...
• 7,538
1 vote
61 views

### Absolute definition of 'True' and 'False' in Mathematical Logic

I searched for the definition of Truth. According to this Wikipedia: https://en.m.wikipedia.org/wiki/Truth , Truth is the property of being in accord with fact or reality. when I searched for the ...
1 vote
34 views

### Showing that $x^3 + y = y^3 + x$ is an equivalence relation

I am asked to prove that: $x^3 + y = y^3 + x$ is an equivalence relation. So far I have the following: Reflexive: $m^3 +m = m^3 +m$ Symmetric: $m^3 + n = n^3 + m \rightarrow n^3 + m = m^3 + n$ ...
16 views

### Boolean algebra simplify with use of complex gates

How to simplify Boolean algebra, And most importantly, "simplify" means at the least use of logic gates | including (nand,nor,xor,exnor...) The Boolean algebra(including K-maps , Quine–...
1 vote
27 views

### How to prove that the derivatives of bent functions are balanced? [closed]

Function $F:\mathbb F_2^m\rightarrow \mathbb F_2^n$ is bent iff $$v \cdot F$$ is bent for all nonzero $v\in \mathbb F_2^n.$ Why is this equivalent to saying $F$ is bent iff $$D_a F = F(x)+F(x+a)$$ is ...
• 49
1 vote
24 views

28 views

### simplification of boolean expression using k-map

I have the following boolean expression which has to be solved using K-map: ...
22 views

### Is every complete Boolean algebra a complete subalgebra of some powerset algebra?

Is every complete Boolean algebra $B$ a complete subalgebra of some powerset algebra $P(X)$, in the sense that the inclusion map preserves infinite meets and joins?
38 views

### Finite state Kleene star machine

I want to represent finite-state machine but I have problems with opening brackets $(a^*dc^* + acd^*)^*$. Should it be $a^*d^*c^* + a^*c^*d^*$? Should I use the first or second image option?
• 1
25 views

### Convert from CNF to DNF

I have $$f(x4) = (x1∨¬x2)*(x2∨¬x3)*(x2∨x4)*(x3∨¬x4)$$ and I need to solve this using distributive law $x(y∨z)=xy∨xz$ and $A*0=0,\;$ $A∨0=A$ and $A∨A*B=A.$ but I can't understand the last line of the ...
• 1
35 views

### Extending the inclusion map of Boolean algebras using Sikorski's extension theorem (Exercise 7.31 of Jech)

Exercise 7.31 of Jech's Set Theory says: If $B$ is a Boolean algebra and $A$ is a regular subalgebra of $B$ then the inclusion mapping extends to a (unique) complete embedding of the completion of $A$...
• 7,538
38 views

### Karnaugh map and boolean simplification yield unsatisfactory results.[Solved]

My binary logic for my circuit is \begin{array}{|c|c|c|c|c|c|} A &B &C &D &Hallway &Stairs\\\hline 0 &0 &0 &0 &0 &0\\ 0 &0 &0 &1 &1 &1\\ 0 &...
1 vote
36 views

### How is it the case that: Any complete lattice is a Boolean algebra.

In the book “A Functorial Model Theory” by Nourani (pg152), it is stated that However, I didn’t understand what does he mean? Because a complete lattice is not even necessarily distributive whereas ...
• 187
47 views

### Is it possible to convert boolean operations to addition/multiplication modulo $2$?

I mean, is it possible to convert AND OR NOT to simple algebraic expressions? If it's not ...
13 views

### Why are the row space and column space lattices of a boolean matrix inverses of each other?

We define the row/column space of a matrix $A$ to be all possible sums of rows/columns of $A$. The row space ($V(A)$) and column space ($W(A)$) form a lattice. See example matrix and respective row ...
137 views

### Clarification of the definition of a clone

I've begun studying boolean algebra in mathematical logic, after studying a course on abstract algebra which ended with the definition of boolean lattices, so I'm familiar with some algebraic ...
• 1,121
47 views

### How to proof the property of cofactor?

I'm studying formal verification, and I am faced with the problem: Let $f$, $g$ be Boolean functions. Proof or disproof that $$\neg(f_v) = (\neg f)_v$$ $$(f ⊕ g)_v = (f_v ⊕ g_v)$$ where $f_v$ denote ...
• 15
20 views

### Generating Monotone Boolean Function

I recently went thru an article over Generating Monotone Boolean Function (at https://www.mathpages.com/home/kmath094/kmath094.htm ) but couldn't understand the concept of using 2 monotone functions ...
• 1
1 vote
40 views

### $FA_{\kappa}(\mathbb{P}) \iff BFA_{\kappa}(\mathbb{P})$, if $\mathbb{P}$ is a $\kappa^+$-cc poset.

I'm trying to prove the question in the title, where the axioms are enunciated like $FA_{\kappa}(\mathbb{P})$ is the assertion that for each colection $\{I_{\alpha} : \alpha < \kappa\}$ of maximal ...
• 1,254
25 views

### Question on Quotient algebra

I was wondering if anybody could help me to solve the following problem I have been thinking about. Manny thanks in advance. Consider the quotient algebra $\mathcal{P}(̩\omega)/fin$, where is fin the ...
27 views

### I am missing something important with Boolean Algebra

It seems that almost every Boolean expression I try and solve, I always find a Boolean Discrepancy . I could go back and use a different property, theorem, or law at different points of the process, ...
27 views

### Simplification of sum of products

I have the following equation: A'BC + AB'C + ABC' + ABC I know I can simplify one part of the equation factoring AB(C' + C) = AB I looked at the results in an online solver and the simplification of ...
19 views

• 23.1k
37 views

### $(p \rightarrow q) \land (\lnot p\rightarrow q) \equiv q$

Prove that $(p \rightarrow q) \land (\lnot p\rightarrow q)$ is logical equivalent to $q$ by using a chain of logical equivalences. The question states explicitly not to use a truth table. I tried the ...
• 588
10 views

### Is this expression in a POS form?(doesn't matter if it is in a canonical form)

A'B' + A'CD + A'DE' and I sould convert it to POS, is what I worte below correct?(I already mentioned that it should necessarily be in a canonical form) (A')(B' + CD + DE')
26 views

### Meaning of expression

I am studying boolean algebra, would anybody know what the below symbol before the letter means: Thanks
27 views

### Boolean Algebra - What does this mean

I am reading a book on Boolean Algebra and at one point it had this expression: It said that if a,b (weird E) (weird B), then a or b = b or a Any idea what the "weird" E means and the whole ...
104 views

### Reversing a complex boolean function

I am trying to reverse-engineer the DRAM address function of a memory controller. This is a function that maps a physical address to a DRAM bank. More formally, I am trying to find functions $f_i$ for ...
• 71
35 views

### Bidistributive implies boolean algebra

Honestly I'm not sure if this question is appropriate here so I will likely remove it after I get an answer unless I am told otherwise. This comment on reddit https://www.reddit.com/r/math/comments/...
• 130
1 vote
63 views

### I'm studying boolean algebra in the discrete mathematics

problem Let $a_4a_3a_2a_1a_0$ be a 5- bits binary numbers (each $a_i=0$ or $1$). Write an expression in Boolean algebra that evaluates to $1$ when odd number of bits of the number is $1$ and $0$ ...
• 55
1 vote
32 views

### Generalization of Łoś theorem to infinitary logic

"It is straightforward to check that, essentially by the same proof as for $\mathcal{L}_{\omega \omega}$, Łos’s Theorem 0.6 holds for $\mathcal{L}_{\kappa \kappa }$ and ultraproducts by $\kappa$...
I started to study set theory; my question is about definition of filters in Cori-Lascar's book Mathematical Logic. Definition (Filter): A filter in a Boolean Algebra $\mathcal{A}=(A,+,\times,0,1)$ is ...