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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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 $(x^3)(Y) + 2XZ + WX + W$ mod $2$ to an ...
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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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Simplify Boolean Algebra Expresssion q1 [on hold]

X = ABCD +A' X = A' + BCD Can explain why A become A'+BCD?
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11 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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35 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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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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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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21 views

Can you prove the bool Exmpression [closed]

Can you prove the bool Exmpression a'+a'.b+a.c+a.b.c
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27 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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18 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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16 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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1answer
17 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
40 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
38 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 + ...
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1answer
18 views

How can I prove that (x and ¬y) or (¬x and y) = ¬((x and y) or (¬x and ¬y))?

I'm stuck at this problem: (x and ¬y) or (¬x and y) = ¬((x and y) or (¬x and ¬y)) Basically what I have to do is to convert the right side of the equation to the left side using boolean algebra. I ...
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18 views

How to prove that $abd = abcd + abc'd$ for all general occassions

It is true for example that $abd = abcd + abc'd$. Each of the terms on the right part of the equation contains all the used letters. Is there anyway to prove that any term is equal to the sum of the ...
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1answer
22 views

Simplifying boolean algebra expression $(AB+AC)'+A'B'C$

$$\eqalign{(AB+AC)'+A'B'C&=\overline{(AB+AC)}+\overline A \,\overline BC\\&=(\overline A+\overline B)(\overline A+\overline C)+\overline A\,\overline BC\\&=\overline A+\overline ...
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Boolean algebra-dual of an expression

Can anyone think of an expression that is equal to its dual ? I've been trying to solve this for the past 2 hours, but nothing comes to mind.
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1answer
40 views

Proof for $\forall x A \Leftrightarrow \neg \exists x \neg A$

I try to proof, that $\forall x A \Leftrightarrow \neg \exists x \neg A$ I know how to proof, that $\forall x A \Leftrightarrow \exists xA$, but I don't understand, how to get negation.
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3answers
32 views

Implementing logic functions using only an OR gate with one input inverted

I've been looking at logic gates, boolean expressions and Karnaugh maps. I ran into a question regarding whether it was possible to implement all logic functions using only one logic gate: an OR gate ...
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32 views

How to simplify the Boolean function $A'B'C + A'BC' + ABC + AB'C'$?

So the question I have asks to implement the circuit with $XOR$ gates. So I am 3/4 through the problem when I am having problems simplifying the Boolean expressions below: $$A'B'C + A'BC' + ABC + ...
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1answer
21 views

Boolean algebra - neutral elements

I am searching for the neutral elements of following Boolean expressions: -NOT -NAND -NOR The neutral element of NOR should be 0 (false) but the others? I think for NOT and NAND there are no neutral ...
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9 views

Simplify the Boolean functions using K-Map

I was able to derive these boolean expressions correctly from a circuit diagram. (Professor put answers up to compare) She now wants us to use a K-Map to simplify these functions. This where I am ...
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1answer
100 views

Is infinite boolean algebra atomless?

I got two questions: 1) Does there exist an infinite Boolean algebra which contains an atom? I answered yes. 2) Does there exist an infinite Boolean algebra B such that for every b contained in B ...
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1answer
121 views

Finding the atoms and elements of a Lindenbaum–Tarski algebra

Let B be the Lindenbaum–Tarski algebra with three variables $p,q,r$ (1) Find all the atoms of $B$. (2) How many elements of does $B$ have? So I think I know what an atom is, but I'm still not sure ...
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15 views

How to simplify the given boolean expression to simplest form? [duplicate]

I have the expression xy+xy'z+x'yz'. I have tried a number of ways to simplify it. What approach will ensure that this expression is reduced to its simplest form?
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question in math logic: find the d.n.f. and c.n.f.

The question is as follows: Find the disjunctive and conjunctive normal forms of the following: $$ (A \to (B \to C)) \to ((A \to \neg C) \to (A \to \neg B)) $$ My solution is as follows, but I ...
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26 views

How to proof tautology without truth table in this case? [closed]

Hej, i got stucked while finding a solution to proof the following is a tautology. Can someone help me out please with a good tip? Thanks in advance
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1answer
25 views

simplify boolean expression: xy + xy'z + x'yz'

As stated in the title, I'm trying to simplify the following expression: $xy + xy'z + x'yz'$ I've only gotten as far as step 3: $xy + xy'z + x'yz'$ $=x(y+y’z) + x’(yz’)$ $=x(y+y’z)+x(y’+z)$ But I ...
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1answer
19 views

Boolean Algebra, using DeMorgan's law

I have obtained this function: $$(\overline{A}*D) + (\overline{A}*C) + (\overline{B}*\overline{D})$$ ... after I have used Karnaugh Map to simplify the canonical expression. And now, I am needing ...
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12 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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28 views

Is there a connection between Boolean algebra and probability?

Is there a unifying abstraction that links Boolean algebra and probability theory? Both Boolean algebra and probability provide us the means to answer questions about set participation. On the one ...
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How to express other logical operations via Pierce's arrow?

x↑y, x⇒y, and x⇔y. So I have really given my best, but all I could do is express the conjunction, disjunction, negation, and impilcation.
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Defining an example of a Boolean algebra (Discrete Math)

This question is listed in my textbook: Give an example of a Boolean algebra B and elements $x$, $y$, $z$ in $B$ such that $x + z = y + z$, but $x \neq y$. Now, I believe this means I have to ...
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19 views

Simplify (x'+y)'(x+y)' with boolean algebra

So I'm doing some homework and trying to simplify (x'+y)'(x+y)'. So far these are the steps I've completed, but I'm not 100% sure that they're appropriate. $(x'+y)'(x+y)' = (x'+y)'(x’y’)$ ...
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12 views

Help with boolean algebra simplification

How can I simplify this using Boolean algebra: wx'z+xy'z I was thinking of distributing the z so it would be z(wx'+xy') but what should I do next? Help will be greatly appreciated.
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10 views

Boolean Algebra simplification

How can I simplify this expression? given: wx'z+y'z'+xz'+xy'z my work: z(wx'+xy')+y'z'+xz'is this step correct? How can I simplify further, maybe to three terms?
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1answer
25 views

Karnaugh map minimal representation

Find the minimal representation for: f(w,x,y,z)= summation m(0,5,6,8,13.14)+d(4,9,11,12) I was a little confused what to do with the don't cares but I used all of them.. Based on the Karnaugh map ...
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55 views

Can this expression $(\neg B \land \neg D) \lor (\neg A \land B \land C) \lor (A \land C \land D)$ be further simplified?

I have assignment for computer architecture where I have to simplify a big boolean function: f(a, b, c, d) = a'b'c'd + a'bcd' + abcd + a'bcd + a'b'cd' + ab'cd' + ab'c'd' + ab'cd + a'b'c'd' ...
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30 views

Show that a interval from a boolean algebra is also a boolean algebra and that a function is surjective

We have an boolean algebra $(B,\lor, \land, ', 0, 1)$ and $b \in B - \{0\}$. We consider $[0,b] = \{x \in B | 0\le x\le b \} \subset B$, where $\le$ means an order relationship introduced in the ...
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17 views

Boolean algebra simplification

Is this the simplest form of the expression? Given: $$x'y'z+x'yz'+x'yz$$ My work: $$x'y(z'+z)+x'yz'= x'y+x'yz= x'y(z)$$
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how does $(p\to q)\lor r \lor s$ effect $(p\leftrightarrow q) \lor r \oplus s$

If we know that $\lnot p \lor q \lor r \lor s=\top$, then what is the value of: $(\lnot p \land \lnot q) \lor (p \land q) \lor(r \land \lnot s) \lor (\lnot r \land s)$ I tried doing it with a truth ...
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1answer
25 views

Implement using only XOR and AND gates [closed]

How can I implement the function: F = ABC’D + AD’ + A’D using only XOR and AND gates ?
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27 views

Proving Boolean Logic using Axioms

I am trying to prove boolean logic formulas using axioms. I have a lot of trouble deciding what to do, which axiom to use, when to use it, etc. I've asked others on proving formulas but they have all ...
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What am I missing about this Boolean expression expansion?

Sorry if this is too basic, but I am working through Boolean Algebra and Its Applications and do not understand this expansion in the author's example 5 in section 1-6: $$(A+X+Y)(A+B'+Y') \rightarrow ...
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114 views

Why can't a Venn diagram constitute a proof?

I'm reading Boolean Algebra and Its Applications and come across this statement about Venn diagrams: It should be remembered that such diagrams do not constitute proofs, but rather represent ...
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Boolean Simplification of a large problem

I am unsure where to even start on this problem. My intuition that what ever can be done to the original problem can be done over and over to simplify the whole thing. Please help with some guidance. ...
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1answer
22 views

Boolean Simplification of $ (a+b) \cdot (a \cdot c + a \cdot \overline{c}) + a \cdot b + b $

Below is my simplification, but my truth tables don't line up, but I can't find my error. $ (a+b) \cdot (a \cdot c + a \cdot \overline{c}) + a \cdot b + b $ $ (a+b) \cdot a \cdot (c + \overline{c}) ...