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

learn more… | top users | synonyms

1
vote
1answer
21 views

Can covering be done on two elements?

The covering rule is: $$B \bullet (B+C) = B$$ and $$B+(B \bullet C)=B$$ So does it follow from this rule that: $$B \bullet A \bullet \bar{C} + B \bullet D \bullet\bar{F} = B \bullet ...
1
vote
0answers
14 views

How to solve this boolean algebra problem?

Given two expressions: $$A\bar{D}+A\bar{C}D +A\bar{B}C + ABCD = Y$$ and $$BD+A\bar{C}D=Z$$ is there a way to simplify this using Boolean Algebra? I tried different combinations, but if I factor out AC ...
1
vote
1answer
20 views

Arrow's Impossibility Theorem Using Boolean Algebra

I am currently working on a research project which involves using Boolean matrices for the proof of Arrow's Impossibility Theorem and various other lemmas and results related to quasi ordered sets. In ...
0
votes
0answers
10 views

Boolean Function with ^ and or

Please provide feedback on my answer to this question. Question: Prove that not every boolean function is equal to a boolean function constructed by only using ^ and or. Answer: True, Suppose that a ...
0
votes
2answers
30 views

finding a boolean function with specific property

The problem I am trying to solve is: Prove that not every boolean function is equal to a boolean function constructed by only using $\wedge$ and $\vee$. My solution is $$\left(p\wedge\thicksim ...
2
votes
1answer
25 views

Boolean Least Squares semidefinite relaxation

So I'm working on the Boolean least squares problem that comes up a lot in circuit design. In its raw form, it looks like this, $$\phi = \min \operatorname{trace}(A^TAX) - 2b^TAx + b^Tb$$ s.t. $$X = ...
2
votes
0answers
38 views

Simplify Product of Sums

Similar question to: Boolean Algebra - Product of Sums I was given a truth table and asked to give the sums-of-products and the product-of-sums expressions. I reduced the sums-of-products ...
1
vote
1answer
18 views

Proving Boolean Functions

Please check my answer to this question and give me feedback . Question: Either exhibit 333 different boolean functions on the three variables p,q,r, or prove that there aren't 333 different such ...
0
votes
1answer
18 views

Boolean Functions with p,q,r

Please give me feedback for my answer to this question. Question: (1) Are the boolean functions $(p \land \neg q) \lor ( \neg r \land q)$ and $(p \lor \neg q) \land (r \lor \neg q)$ equal?. Explain ...
2
votes
3answers
48 views

Construct XNOR with only OR gates

Is it possible to construct the XNOR gate which is given as, a XNOR b = (a AND b) OR (~a AND ~b), by using only OR gates. So from the definition, the question boils down to: can you construct the AND ...
0
votes
1answer
21 views

A boolean algebra is complete iff its Stone Space is extremally disconnected

I have the following proof, but I don't understand one of the steps: Theorem 4.4. A Boolean algebra is complete iff its Stone space is exlremally disconnected. Proof. Identify the given ...
0
votes
0answers
14 views

Boolean Algebras and Spaces

Show that a countably infinite free Boolean algebra $B$ has a Boolean space homeomorphic to $2^\omega$; where $2$ is the discrete space $\{0,1\}$; hence B is isomorphic to the Boolean algebra of ...
0
votes
1answer
21 views

showing Boolean algebra equality

I have this exercise in my worksheet : Show that x (z ⊕ y) = xz ⊕ xy I reached this in solving it , but didn't reach the final equation x(z'y + zy') xz'y + xzy' please can someone show how
1
vote
1answer
262 views

Logic Circuits And Equations Issue - Multiply Binary Number By 3

I am trying to build a logic circuit that multiplies any 4 digit binary number by 3. I know that if I multiply\divide number by 2 it moves left\right the digits, but what I`m doing with multiply by ...
1
vote
1answer
22 views

Boolean-like algebra

Suppose one had an algebra that that follows most of the laws of Boolean algebra (associative, commutative, distributive, identity, annihilator, idempotent, double negation, De Morgan) but does not ...
0
votes
1answer
38 views

A well-defined operation on measure algebra

Let $(X,\cal{M},\mu)$ be a measure space, and for $E,F\in \cal{M}$ write $E \sim F$ iff $\mu(E \Delta F)=0$. Let $\widetilde{\cal{M}}$ be the set of equivalence classes in $\cal{M}$ for $\sim$; for ...
1
vote
2answers
33 views

How to prove the divisors of 15 form a Boolean algebra

This from Exercise 3.1 in "A Beginner's Guide to Discrete Mathematics" Let B be the set of all positive integer divisors of 15, that is B = {1, 3, 5, 15}. Prove that B forms a Boolean algebra with ...
3
votes
1answer
62 views

Prove if Tautology, Contradicton, or Neither. Is my proof ok?

Determine whether $((p \Rightarrow q) \Rightarrow r) \Leftrightarrow (p \Rightarrow (q \Rightarrow r))$ is a tautology, a contradiction, or neither. If $p,q,r = (0,0,0)$ then $((p \Rightarrow ...
0
votes
1answer
36 views

Alternative to xor(A,B,C)

How can we make a comprehensive statement, which will correspond to the truth table of xor (A, B, C) by combining logical operators AND (&), OR (|), XOR (xor) and NOT (!)?
0
votes
2answers
2k views

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 ...
1
vote
1answer
28 views

Convert a boolean function into K-map

I would like to know how can I convert the following boolean function into a truth table and accordingly construct the k-map $$F = A'B'C'+B'CD'+A'BCD'+AB'C'$$ thanks in advance :)
1
vote
0answers
30 views

boolean functions on p,q,r

Either exhibit 333 different Boolean functions on the three variables p; q; r, or prove that there aren’t 333 different such functions. the answer i have come up with is: the number functions on n ...
0
votes
0answers
20 views

Homework Question [duplicate]

How to prove that not every boolean function is equal to a boolean function constructed by only using ∧ and ∨. Should I choose a specific function or just compare any two Boolean functions constructed ...
0
votes
0answers
26 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: ...
4
votes
2answers
50 views

Peculiar examples to the Stone Representation Theorem

The Stone Representation theorem states that every Boolean algebra is isomorphic to a field of sets. That is, a Boolean algebra whose elements are sets, and sums, products, negation are union, ...
0
votes
0answers
16 views

algèbre de boole

hello By a communicating channel 0 and 1. As a result of a spurious noise (lightning, an electric switch manipulation, ...) transmitting a 0 is sometimes received as a 1 and vice versa. Let E = ...
0
votes
1answer
18 views

Number of non-increasing boolean functions of $n$ booleans, up to permutations.

How many non-increasing boolean functions of $n$ boolean variables are there? I don't want to count functions that ignore some of their inputs. If two or more functions differ only by permuting their ...
10
votes
0answers
255 views

Is this $really$ a categorical approach to $integration$?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
2
votes
0answers
41 views

Is this a good enough proof?

Is this proof good enough? If not, any feedback would be appreciated. Thanks. Either exhibit $333 $ different boolean functions on the three variables $p; q; r,$ or prove that there aren’t $333$ ...
2
votes
1answer
53 views

How to prove that linear functions cannot represent binary functions

Yesterday, I thought about representing boolean algebra as linear functions: For some vector space $V$ and for some $A, B \subset V$ such that $A \ne \emptyset \,\wedge\, B \ne \emptyset ...
1
vote
5answers
64 views

If one of the hypotheses holds, then one of the conclusions holds. (looking for a proof)

Using a huge truth table, I proved the theorem below. I cannot find a more elegant proof. I tried to rewrite expressions; e.g. using the distributive laws and the laws of absorption - to no avail. Is ...
0
votes
0answers
13 views

Boolean Algebra Simplification help

Can anybody help me solve this Boolean algebra, Im a bit stuck on it and any assiastance would be great. Thanks A'B'C'D' + A'B'CD + A'BCD' + A B'C'D'
2
votes
3answers
53 views

Proving $\neg A\vee(A\wedge \neg B)= \neg A \vee \neg B$.

How do I prove using boolean algebra that $\neg A\vee(A\wedge \neg B)= \neg A \vee \neg B$? I can see it in the logic table and it is logical, but I can't prove it mathematically.
3
votes
1answer
194 views

Solving special boolean equation set

I have boolean equation sets that look like this (where ^ means xor): eq 1: x1^x3^x5^x6^x9^x10^x11^x13^x17^x18 = 0 eq 2: 1^x1^x3^x10^x12^x17 = 0 eq 3: 1^x2^x3^x5^x8^x10^x14^x16 = 0 ...
0
votes
1answer
17 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: ...
0
votes
2answers
268 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 ...
1
vote
1answer
25 views

Number of Positive Definite Binary Matrices

How may positive definite matrices (over finite field- $F_p$) are possible? What is the criterion in getting those?
4
votes
6answers
2k views

how to solve system of linear equations of XOR operation?

how can i solve this set of equations ? to get values of $x,y,z,w$ ? $$\begin{aligned} 1=x \oplus y \oplus z \end{aligned}$$ $$\begin{aligned}1=x \oplus y \oplus w \end{aligned}$$ $$\begin{aligned}0=x ...
1
vote
2answers
49 views

Assignment for discrete mathematics

How can I prove that not every boolean function is equal to a boolean function constructed by only using ∧ and ∨?.Need help in proving it.
0
votes
0answers
10 views

How do Boolean-valued functions work?

Consider this function: P: X→ {true, false} There's nothing in that expression that says when X is true and when it is not true. How do these work?
0
votes
3answers
67 views

Discrete Mathematics (boolean)

Either exhibit 333 different boolean functions on the three variables p; q; r, or prove that there aren’t 333 different such functions $p$ $q$ $r$ $0 0 0$ $001$ $010$ ...
0
votes
1answer
35 views

prove that there does not exists a boolean algebra containing only three element

please prove that there does not exists a Boolean algebra containing only three elements .prove it with example so that i can understand easily.i cant understand the question and i could not tried to ...
0
votes
1answer
37 views

How can I simplify this boolean equation for the multiplexer a little further?

I've obtained a formula through cannonical representation, which is: $$A\cdot \overline{B\cdot S}+A\cdot B\cdot \overline{S}+\overline{A}\cdot B\cdot S+A\cdot B \cdot S$$ And I'm trying to simplify ...
1
vote
0answers
32 views

What are three possible ways to express the following Boolean function with eight or fewer literals?

F= A'BC'D + AB'CD + A'B'C' + ACD' I assumed that the question was asking for me to simplify. I placed the terms into a kmap and have gotten SOP F= A'B'C' + A'C'D + AB'C + ACD' or POS F= ...
3
votes
0answers
25 views

Finding the atoms of a Boolean Algebra

I have a homework question that asks me to find the atoms of the Boolean Algebra that contains 256 Boolean functions "such as F1(x,y,z) = x + y +z, F2(x,y,z) = x + xz, F3(x,y,z) = xyz+ xyz and so on". ...
0
votes
0answers
16 views

Proof by Induction… For any Boolean function F we can define its dual fd by?…

For any Boolean function $f$ we can define its dual $f^d$ by: $ f^d = ( x_{1}, x_{2},...,x_{n}) = \overline f(\bar x_{1}, \bar x_{2},...,\bar x_{n}) $ How do I prove this by induction?
0
votes
1answer
3k views

Convert boolean expression into SOP and POS

Convert the following expression into SOP (sum of products) and POS (product of sums) canonical forms using boolean algebra method: $(ac + b)(a + b'c) + ac$ Attempt at solution: $(ac + b)(a + b'c) ...
1
vote
0answers
10 views

Logic subject-reductio ad absurdum

Can you solve this using method reductio ad absurdum? 1)A ↔ (¬ B v C) ¬ A ¬ B 2)¬(R∧ (S v T)) 3)R∧¬ T S ¬R∧ S
0
votes
0answers
18 views

Simple explanation of a Boolean function?

I took on the challenge to self study discrete math and I've come to Boolean functions. Please note, that I'm new to set notation (just learned it) and the form of the Boolean function confuses me ...
0
votes
0answers
9 views

Boolean prime covering

Let $\mathbb B^n$ be an n-dimentional boolean cube. The set ${E}$ of edges is called its 1-cover if any vertex of $\mathbb B^n$ belongs to exactly one edge from ${E}$. The 1-cover is prime if no ...