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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Simplifying P AND (P OR NOT Q)

How can I simplify this? I've tried invoking Demorgan's Law and I get P AND (NOT (NOT P AND Q)) but I can't seem to simplify this further. The answer is P, but how can I prove this?
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1answer
53 views

Stone Representation Theorem

Given two Boolean algebras $A$ and $B$ such that $A$ is a subalgebra of $B$. What is the relation between the Stone space of $A$ and the Stone space of $B$. The question maybe silly but I am getting ...
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Boolean algebra and boolean subalgebra

I have to prove that set of all dividers of number 210 with appropriate operations forms a Boolean algebra. And describe these operations and create a Hasse diagram. In the secondd part I have to ...
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HOW TO CONVERT SOP TO POS [on hold]

How to convert sop to pos ABC+AB'C'+AB'C+ABC'+A'B'C
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1answer
32 views

An example of an ultrafilter

This is Theorem 3.5 (pp. 150) of the book "A course in universal algebra." http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf Theorem 3.15. Let $\bf B$ be a Boolean algebra. (a) ...
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3answers
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De-Morgan's theorem for 3 variables?

The most relative that I found on Google for de morgan's 3 variable was: (ABC)' = A' + B' + C'. I didn't find the answer for my question, therefore I'll ask here: ...
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2answers
462 views

calculating number of boolean functions

I would just like to clarify if I am on the right track or not; I have these questions: Consider the Boolean functions $f(x,y,z)$ in three variables such that the table of values of $f$ contains ...
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1answer
20 views

Trying to simplify boolean algebra a+ac+ab

I am trying to simplify A+AC+AB. I think I have solved it, but I want to double check its right, can it be simplified to A+A(C+b) and then again to A(C+B) as A+A = A?
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1answer
18 views

DeMorgan's Law with Boolean Algebra

So I'm studying for an Assembly Language final tomorrow and I'm trying to simplify the following expression using Boolean Algebra. Here are the steps I've written so far, am I safe in assuming that ...
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1answer
27 views

Getting sum of products from products of sum

I need to write the following Boolean expression in the form of sum of products $F(A,B,C,D)= (A+B+C+D)(A'+B'+C+D')(A'+C)(A+D)(B+C+D)$ I just want to know how to deal with the missing letters. Is $ ...
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1answer
21 views

Write the following Boolean expression in product of sums form?

Write the following Boolean expression in product of sums form: a'b + a'c' + abc is it correct if I write it as the following ? (a+b')(a+c)(a'+b'+c')
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1answer
106 views

Boolean algebra - cube - minimal disjunctive normal form

I have a test coming up and I would like to know how to solve these kinds of problems. This is the description: ...
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1answer
50 views

Boolean algebra: Minimizing a product of sums expression?

For the life of me, I can't figure out how to get this into minimal product of sums form. Any help is appreciated. (a+b+c)(a+b'+c)(a+b'+c')(a'+b'+c')
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1answer
14 views

Simplifying Boolean Function with Karnaugh Map

How to write Product-of-sum(POS) and Sum-of-product(SOP) Above K-Map? I already write POS please check my answer.
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1answer
30 views

Simplying Boolean-Logic Expression

Can you help me simplify this or is this the simplified form? A = (X + Y + Z) (X + ~Y + ~Z) (~X + Y + ~Z) (~X + ~Y + Z) Here's my attempt: ...
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1answer
29 views

How many satisfying assignments are there in a set of 3-CNF clauses where no clause share the same variable?

Say I have a set of 3-CNF clauses $$\mathcal{S} = \{ x_1 \vee x_2 \vee \bar{x_3}, ~~x_4 \vee x_5 \vee x_6\}$$ where $\bar{x}$ is the negation of $x$. Each variable range over $\mathbb{Z}^2$. How ...
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1answer
584 views

Simplify the following expression using Boolean Algebra into sum-of-products (SOP) expressions

Simplify the following expression using Boolean Algebra into sum-of-products (SOP) expressions $Q.S.U + (Q' + S').(R + V) + U.(R + V) + Q' + S.T.U$ $.$ = AND $+$ = OR This is what I have so far ...
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1answer
42 views

Logic expression simplification

I want to simplify this logic expression: Y = (A ∧ B ∧ ¬C ∧ D ) ∨ (C ∧ ¬D) ∨ (A ∧ B ∧ C) ∨ (¬A ∧ C) I know it must become Y = (A ∧ B ∧ D) ∨ (C ∧ ¬D) ∨ (¬A ∧ C) and I found it with Karnaugh, but I ...
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1answer
37 views

Jayne's Equation 1.13 Derivation

Dear Stack Exchange Members, I'm reading 'Probability Theory - The Logic of of Science" by ET Jaynes, and I'm on pg. 11. Jayne's says: *"...For example, we shall presently have use for a rather ...
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1answer
14 views

Simplification of expressions?

The expression below fd < S && ld > e || fs > s && ld > e || fd > s && ld < e || fd < s && ld < e Is the ...
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2answers
940 views

problem simplifying boolean algebra expression using consensus theorem

Please simplify this logic expression for me with helping boolean algebra : A'C'D + A'BD + BCD + ABC + ACD' I know that must use consensus theorem . my solve : STEP 1 : Terms 1 & 3 ...
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1answer
21 views

Complete subalgebra of regular open Boolean algebra generated from open intervals

Let $X$ be a totally ordered set, considered as a topological space with the order topology. The regular open subsets of $X$ (i.e., the sets $U = \operatorname{int} \operatorname{cl} U$) form a ...
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2answers
34 views

Boolean Algebra: Simplifying product of sums

I'm trying to simplify (A+B+C)(A+notB+C)(notA+B+notC) The K-map gives me (A+C)(notA+B+notC) but when I use boolean ...
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0answers
7 views

Converting a boolean expression into CNF and DNF

Is there any systematic way to convert the following boolean expression (QUBO) into CNF or DNF? Here, $x_1, \ldots, x_n \in \{0, 1\}$, $a_1, \ldots, a_n \in \mathbb{Z}$ and $b_{1,1}, \ldots, b_{n,n} ...
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1answer
287 views

Convert expression to NAND only

Endless youtube videos and reading through notes later I am yet again stuck. I have to covert the following to NAND only $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot ...
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2answers
36 views

How to efficiently determine if any two propositional formulas are equivalent

Given any two arbitrary propositional formulas (but only using $\land, \lor, \lnot$), like $\lnot(A \land (B \lor \lnot B) \land C)$ and $\lnot C \lor \lnot A$, how can I (or a computer) efficiently ...
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2answers
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self dual boolean function

How many self-dual Boolean functions of n variables are there?Please help me how to calculate such like problems. A Boolean function $f_1^D$ is said to be the dual of another Boolean function ...
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39 views

Knights and Knaves island [duplicate]

You appear on the Island of Knights and Knaves. Knights always tell truth, knaves always lie. You meat three inhabitants, Carl, Peggy and Zippy, and hear the following conversation: Carl says, "I ...
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2answers
17 views

What is $A^c \cap B^c \cap C^c$

I am working with boolean algebra for my Navy coursework and I was wondering if anyone knew what the formula for $A^c \cap B^c \cap C^c$ is? Also does $A^c \cap B^c \cap C^c = (A \cap B \cap C)^c$? ...
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0answers
19 views

What is the following boolean equation: ΣM(1,2,4,7)?

I am supposed to find the minimum-cost SoP form of that equation Σm(1, 2, 4, 7) for a homework question. However, this is extremely unclear to me. As I understand it this is the summation of the ...
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1answer
26 views

How many n-ary Boolean functions essentially dependent on each of their arguments?

How many n-ary Boolean functions essentially dependent on each of their arguments? essentially dependent means that $$f(b_1,…,b_{i−1},0,b_{i+1},…,b_n) \neq f(b_1,…,b_{i−1},1,b_{i+1},…,b_n)$$
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How to convert a mod 2 function to an expression in Boolean Algebra

I'm not sure if this is the right place to post it but I have a question I'm having a hard time understanding. The questions is: Convert the function $X^3Y + 2XZ + WX + W$ mod $2$ to an expression ...
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0answers
15 views

Are epimorphisms (defined via an obvious action) of free Boolean algebras whose set of generators is a group automorphisms?

Let $G$ be a group. Consider $B$, the free Boolean algebra with generating set (I'll call them "variables") $G$. Let $F$ be some formula (that is, some fixed element of $B$). Define an endomorphism ...
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Number of elements in a Boolean algebra

Consider a set $X$ consisting of $n$ elements Does the Boolean algebra of all subsets of $X$ (i.e. the power set of $X$) have $2^n$ or $2^{2^n}$ elements? I came across both answers, which confuses ...
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2answers
399 views

Writing a boolean formula and logic circuit that computes mux

Let $mux(p_{11}, p_{10}, p_{01}, p_{00}, x_1, x_0) = P_{x1x0}$ (with all variables bits). Write a boolean formula, and then draw a circuit, that computes mux. For ...
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1answer
36 views

Proof for $∃xA⇔¬∀x¬A$

I want to prove, that $∃xA⇔¬∀x¬A$, using classic axioms. I think, I have to start with the following step: $∃xA⇔∃x¬¬A$ But I do not know, how to make this step, using axioms: $∃x¬¬A⇔¬∀x¬A$
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1answer
260 views

Boolean algebra simplification question

I'm trying to simplify the follow SOP expression: $\bar{A}$$\bar{B}$$\bar{C}$ + $\bar{A}$B$\bar{C}$ + $\bar{A}$BC + AB$\bar{C}$ Using a K-map (unless I've erred) it should simplify to: ...
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3answers
41 views

How to show that if $\neg b = a \land d$ then $a \land \neg b = \neg b$ and $b \land \neg a = \neg a$

I'm new to boolean algebra and am having trouble proving the following simple theorem. Many thanks for any help. If $\neg b = a \land d$ then $a \land \neg b = \neg b$ and $b \land \neg a = \neg a$. ...
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1answer
13 views

Simplifying a Boolean Expression 2

The boolean expression is as follows: (¬A^¬B^¬C)∨(A^¬B^C)∨(A^B^¬C)∨(A^B^C) I have found that A⊕(¬B^¬C) is equal to the above but I have absolutely no idea on how to get this result, I have spent ...
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4answers
16k views

how to make a truth table from an boolean expression

I am trying to make a truth table from an SOP boolean algebra expression. I understand AND, OR, NOT truth tables. I just dont understand these types of tables and their outputs. This is the ...
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5answers
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Duality principle in boolean algebra

All the definitions I came across so far stated, that if a statement is true, then also its dual statement is true and this dual statement is obtained by changing + ...
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1answer
19 views

Product of maxterms

Please help me break the ice in understanding how we derive a product of maxterms, say, for: $xy+x'z $ I could be missing some concept here in this but be patient with me. I have also done SOP and ...
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2answers
29 views

Inequality with respect to transitivity

Given a relation R, R is said to be transitive if aRb ∧ bRc, then aRc. The unequal relation (≠) is not transitive, for instance a≠b ∧ b≠c, then a≠c is an invalid consequent of the antecedent (a≠b ∧ ...
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Is XOR a combination of AND and NOT operators?

I'm not sure whether this is the best place to ask this, but is the XOR binary operator a combination of AND+NOT operators?
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2answers
19 views

Joins in lattices and sublattices

Let $A$ be a lattice, and $B$ be a sublattice of $A$. Why is the join of $A$ included in the join of $B$? That is, why is $\bigcup_{t\in T}^{A} a_t\leq\bigcup_{t\in T}^{B} a_t$? (I am tempted to ...
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1answer
1k views

Simplify Boolean Product of Sums Function

I've got a product of sums expression: F=(A'+B+C')&(A+D')(C+D') I need to show it as a sum of products and then simplify it. Right now I got: F=(A'&D')+(A&B&C)+(B&D')+(C&D') ...
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1answer
41 views

Least and greatest element of the $(\mathbb{N}, |)$

Consider the relation | on $\mathbb{N}$, where $\mathbb{N} = \{0,1,2,... \}$ and $n|m$ means $n$ divides $m$. I know that the pair $(\mathbb{N}, |)$ is a partial order, : (1) Find the least and ...
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1answer
18 views

Is there any way to simplify the following boolean expression?

I was trying to manipulate with litarals and minterms of this booleans expression but it really did not lead to anything that could simplify the expression further.. Not sure if I am doing it wrong or ...
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2answers
17 views

Proving relation in boolean algebra, need help

Here is the logic equation and I am trying to prove the relation ($'$ stands for complement): $$𝑥_1𝑥_3' + 𝑥_2'𝑥_3' +𝑥_1𝑥_3 +𝑥_2'𝑥_3 = 𝑥_1'𝑥_2' + 𝑥_1𝑥_2 + 𝑥_1𝑥_2'$$ What I am doing: ...
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1answer
41 views

Duality Principle in Boolean Algebra - Why do I alway get !F instead of F?

I have the function: F = !(a && d || b || c) Now i apply the duality principle and exchange all * with + ...