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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Properties of distributive lattices and congruences.

Let $L$ be a lattice and let $a,b,c,d \in L$. Show that: $\theta(a,b) \subseteq \theta(c,d)$ iff $\langle a,b\rangle \in \theta(c,d)$ $\theta(a,b)=\theta(a \wedge b, a \vee b)$ Where $\theta$ is ...
4
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1answer
26 views

$\land,\lor$ and $\lnot$ determinate a functionally complete basis

I read that a Boolean algebra is defined by the binary operations $\land$ and $\lor$ and the unary operation $\lnot$ on a set such that $$\varphi\land(\psi\land \chi)=(\varphi\land \psi)\land ...
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1answer
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I have this circuit and I need to use boolean laws to simplify this circuit

I need to use boolean laws to simplify the following circuits need to simplify this so that they contain at least amount of gates: a) (A+B)(C+D)+(A+B)(C'+D')= what I did for a) ...
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3answers
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Propositional Logic : Why is ((p $\Rightarrow$ r) $\land$ (q $\Rightarrow$ r)) $\equiv$ ((p $\lor$ q) $\Rightarrow$ r)

I was working my way through some Propositional Logic and had the following doubt : Why is this true : ((p $\Rightarrow$ r) $\land$ (q $\Rightarrow$ r)) $\equiv$ ((p $\lor$ q) $\Rightarrow$ ...
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The congruence class of a distributive lattice

I need to prove the next thing: For every non-empty ideal $I$ of a lattice $L$ consider the relation $\theta(I)$ defined by: $ \theta(I) = \{⟨a, b⟩ : ( \exists c \in I) a \vee c = b \vee c\}$ Prove ...
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2answers
35 views

how to prove boolean identities

I'm working on 2 boolean proofs (¬p⊕q)=(p⊕¬q=¬(p⊕q) <- I assume its equality law i'm not sure how to do this problem(I verified using truth table but I need to do algebraically) ...
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2answers
22 views

Boolean Algebra simplify

The question is to simplify $$xy'z+wxy'z'+wxy+w'x'y'z'+w'x'yz'$$ Using K-map, the answer is $wx + w'x'z' + xy'z$ However, the question wants me to simplify algebraically, stating laws beside. I ...
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2answers
17 views

If the following statements in which a, b, c,d are involved are simultaneously true, find the values of a-d

Can you please help me solve this ? This exercise says that we have the following statements: $$\lnot a \rightarrow b\tag{1}$$ $$\lnot a \Leftrightarrow c\tag{2}$$ $$\lnot b \rightarrow d\tag {3}$$ ...
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1answer
32 views

Exclusive or (XOR) proof [duplicate]

The question is to prove: X'⊕ Y = X⊕Y' = (X⊕Y)' State laws used (' meaning negation) Thank You
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1answer
26 views

How to get from the statement $(AB'+C'A'+C'B')$ to equivalent statement $(AB'+C'A')$?

I've been working a Boolean algebra problem for probably 2 hours at this point, and while I arrive at a much simplified equivalent expression, there's a simpler one yet. Basically, I start out with a ...
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25 views

What is the universe of a sub algebra generated by $ \{(a \wedge b)\vee(c \wedge b') : a,c \in C\}$ ?

I need to prove the next thing, Let $B$ be a Boolean algebra and $C$ a proper subalgebra of $B$. Let $b ∈ B−C$. Prove that the set $ \{(a \wedge b)\vee(c \wedge b') : a,c \in C\}$ is the universe of ...
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0answers
19 views

Distributivity of lattices and prime ideals.

I need help proving that, if $L$ is a lattice with the property that for every nonempty proper filter $F$ and every ideal $I$ such that $F \bigcap I = \emptyset$ then there is a prime filter $G$ such ...
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1answer
18 views

If a set X has the finite meet property, then there is an ultrafilter such that X is a subset of it.

I need to prove that if $X \subseteq B$ is a set with the finite meet property, then there exists an unltrafilter $U$ of $B$ such that $X \subseteq U$. I know that the finite meet property means that ...
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28 views

Does the class of all periodic subsets of $\mathbb{Z}$ of peroid greater than $k$ form a field of sets?

We say that a subset $X\subseteq \mathbb{Z}$ is a periodic subset of $\mathbb{Z}$ of period $k$ if the set obtained from $X$ by adding $k$ to each element of $X$ is $X$ itself. Does the class of all ...
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$p\implies q = p'\vee q$ and duality

I'm reading Halmos's Lectures on Boolean Algebras. The title is a definition and he then also defines $p\iff q= (p\implies q)\wedge (q\implies p)$. Then the following: The source of these ...
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For every congruence of B, any equivalence class determines completely the congruence.

I am trying to prove that for every congruence $\theta$ of a boolean algebra $B$, any equivalence class determines completely the congruence $\theta$. My strategy is to prove first that the ...
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Problem in the correspondence between boolean rings and boolean algebra through characteristic functions

I was working on the relation between boolean algebras and boolean ring and that they are in fact, the same object. But I find something which seems to be incorrect, It's quite long and I try to give ...
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1answer
67 views

Discrete Math Predicate Logic

Consider truth assignments involving only the propositional variables $x_0, x_1, x_2, x_3$ and $y_0, y_1, y_2, y_3$. Every such truth assignment gives a value of $1$ (representing true) or ...
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boolean simplification , help please [closed]

If we begin with $\;\bar A\,\bar C+ \bar B\,\bar C + A\, B\;$ how can we transform to $\;\bar B \, \bar C + B \,\bar C + A\, B\;$. I'm so lost please help.
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1answer
12 views

Set of Numbers when added in any combination always produce unique result

What I'm looking for is a set of numbers that when added in any combination they always have a unique sum? Is this called something? I have searched on google for hours and I'm having a hard time ...
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1answer
21 views

A question about truth tables

Hello guys i have a question, I am trying to make a truth table which consists out of 4 variables F(A,B,C,D) = B'D + A'D + BD Is it true on the truth table when for example in B'D we have 0001 or ...
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What is the full proof of the theorem - every truth table can be expressed by CNF expression ?

I tried to search (in Google) the full proof , but nothing has been found. Thanks.
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1answer
41 views

Boolean algebra Simplification of “xy + x'z + yz” [closed]

I'd like to simplify the following expression "xy + x'z + yz": ...
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9 views

Homomorphism from a four-element Boolean algebra

I have a set like: $$ S = \{0, a, b, 1\} $$ I need to show all homomorphisms from a four-element Boolean algebra to another four-element Boolean algebra. How to find and write them?
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25 views

All subalgebras of eight-element Boolean algebra

Let's assume that we have a set: $$ X = \{a, b, c\} $$ Is it true, that a Boolean algebra of this set is like below? $$ P(X) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, ...
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1answer
21 views

Number of subalgebras of the power set algebra

Let $X=\{a,b,c\}$ and $\mathcal{P}X=\{\emptyset,X,\{a\},\{b\},\{c\},\{a,b\}.\{a,c\},\{b,c\}\}$. I can only see 4 subalgebras of $\mathcal{P}X$, namely: $\mathcal{F}_0=\{\emptyset,X\}$ ...
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40 views

Stone space of finite Boolean algebras

Is the Stone space of every finite Boolean algebra a finite discrete space (for every finite Boolean algebra is complete, atomic, and isomorphic to the power set of its atoms; and finite discrete ...
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28 views

Trying to prove Equivalency using Boolean Algebra

The question presented was to use boolean algebra to show that XY’Z + X’Y’Z’ + XY’Z’ + X’YZ’ ≡ XYZ’ + XY’Z + XY’Z’ + XYZ’ I've tried using various laws of Boolean algebra, but the answer that I ...
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Automorphism groups of Boolean algebras and atomicity

Let $A$ be a complete Boolean algebra, $B$ a complete Boolean subalgebra of $A$, $G$ a group of automorphisms of $A$. Finally let $Fix_G(A)$ be the subalgebra of $A$ that is fixed by every ...
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Connection between Directed Acyclic Graphs and Boolean Functions

I am given a set of $n$ vertices and testing some properties over the set of all directed graphs over them (i.e. acyclicity and bipolarity). I already done this by generating every undirected graph ...
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2answers
64 views

Why cant AND and NOT represented only with IMPLICATION?

Can someone please explain why can't I use only implication to represent AND and NOT with proof as well? I know that I can represent OR simply by using implication. Was thinking if I could find ...
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Axioms as recreational mathematics

Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the ...
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1answer
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$f(x) = x$ or a , if $f(x)$ and $a$ is known find $x$ boolean algebra

I am new to boolean algebra. I am facing difficulty solving this problem: Given $f(x) = x \lor a$, for some $f(x)$ and $a$, deduce the value of $x$. Can someone provide me the solution with ...
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3answers
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Designing a circuit to verify operation of an OR gate.

Consider the following image: I need to design a circuit that verifies the logical operation of the OR gate. In the above image, the LED will be on (f = 1) if the or gate is working properly. I can ...
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1answer
47 views

Are Boo and BooRng really isomorphic?

In ACC on the top of page 34, I read: The construct Boo of boolean algebras is isomorphic to the construct BooRng of boolean rings and ring homomorphisms. This contradicts my intuition since ...
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1answer
36 views

AND, OR, NOT, and creating turing complete programming languages

Suppose I have an arbitrary computing language, and the following holds: Let all constants be finite, and assume we are computing in binary. An arbitrary number of inputs, A, and outputs, B, can be ...
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1answer
27 views

Boolean algebra proof and cancellation law

I have a Boolean algebra with some elements $a,b,c$. I have to show this: $(a ∧ b) ∨ (a′ ∧ c) ∨ (b ∧ c) = (a ∧ b) ∨ (a′ ∧ c)$. Now I have done other such proofs before but this one I got lost in. I ...
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2answers
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DNF Form of XOR Operator with N Arguments

I’m working on this problem: Explain how to express $p$ using the boolean connectives AND, OR, and NOT so that the resulting expression has length polynomial in $n$. $$p(x_1\cdots x_n) = x_1 \oplus ...
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Boolean algebra gives rise to a ring

There is a well-known result that every boolean algebra $(R,0,1,\land,\lor,\neg)$ can be made a boolean ring by defining $$x\cdot y=x\land y\qquad\text{and}\qquad x+y=(x\land\neg y)\lor(y\land\neg ...
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0answers
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consider a base 16 adder how to modify the adder so that it can perform a base 8 addition

Consider a base $16$ adder. How can I modify the adder so that it can perform a base $8$ addition? I expect this question will appear in my exam tomorrow; if anyone can give me a hint or a solution, ...
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unique children of a point in a boolean lattice

I am working with two-element boolean algebra, e.g. points composed of strings of $0$s and $1$s and bit-wise $AND$ and $OR$ to find maxima and minima. In the domain I'm working in, I need to assign ...
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1answer
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Convergence of monotone boolean network in the worst case

I'm looking for (upper bound) convergence of increasing monotone boolean network (network composed only with OR, AND, identity ($f_i(x)=x_j$) functions) in asynchronous updating mode. It means that if ...
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Converting boolean expression - POS to SOP

Convert the following to sum of products form: (a' + c)(a' + b + c')(a + b') I did the following: multiply out the first two expressions: = (a'a' + a'b + a'c' + ca' + cb + cc')(a + b') ...
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1answer
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Number of bit operations in nxn zero-one matrix boolean product

I was reading transitivity closure from the book Discrete Mathematics and Its Application by Kenneth Rosen It says that in the boolean product of nxn zero-one matrix, there are $n^2(2n-1)$ bit ...
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Question on support of measure

Let $(X,\Sigma,\tau,\mu)$ be a topological measure space, and let $K=\text{supp}(\mu)$ and $\lambda=\mu|_K$. If $M=\Sigma/N$ and $L=\Sigma_K/(N\cap K)$, where $N$ is the set of al nulls. As I think, ...
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What is meant by $AB$ in boolean algebra?

I am endeavoring to teach myself Boolean Algebra. Oh what fun! From the questions I've read on this site, one of the most common notations I've seen is $AB$ (examples: here, here, and here). Problem ...
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1answer
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consider a base-16 adder. explain how to modify the adder so that it can perform a base-10 addition

consider a base-16 adder. explain how to modify the adder so that it can perform a base-10 addition I found this when I searched in Google but not understand please guide me to understand this ...
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39 views

How to simplify Boolean expression: $(C'B')+(CB)$

I'm very weak in math and logic, and currently tried doing K-map, and got this as result: $$(C'B')+(CB)$$ My question is, can this be further simplified? I tried it myself, but I got $0$ (False). ...
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1answer
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Boolean algebra with measures

Let $A,B$ be two Boolean algebra with measures $m,p$ thereon, respectively such that the measure algebra $(A,m)$ is isomorphic to the measure algebra $(B,p)$. Suppose that we have two isomorphic ...
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217 views

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?