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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Boolean Algebra Minimization

Prove that $\bar{A}B + AC + BC = \bar{A}B + AC$ with the help of boolean algebraic manipulations. I have no clue from where to start, how should I tackle these type of questions? Or $$ ...
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Question on homogeneous algebra

Defn: A Boolean algebra $B$ is homogeneous if for every non-zero $a\in B$, $B$ is isomorphic to $B|_a$. e.g. the algebra $\mathcal L$ of all Lebesgue measurable sets in $[0,1]$ modulo null sets. ...
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Countable additive of a measure

Suppose we have a field of sets $\mathcal F$ such that no infinite union of members of $\mathcal F$ belong to it. Let $m$ be any finitely additive measure on $\mathcal F$, then $m$ is ...
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How would I go from DNF to a simplified formula with less symbols?

Here's a DNF: $$(\neg A_1 \land \neg A_2 \land \neg A_3 ) \lor (A_1 \land \neg A_2 \land \neg A_3 ) \lor (\neg A_1 \land \neg A_2 \land A_3 ) \lor (\neg A_1 \land A_2 \land \neg A_3 )$$ And the ...
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1answer
23 views

Boolean Algebra: making a proof assistance

So far i've tried all the identities my teacher gave us and keep getting stuck I have to prove that x'y' + y = x' + xy using boolean algebra identities
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33 views

Axiomatic proof and Boolean algebra?

I'm trying to prove that: $$(c'd') + (bc') + (a'b'c) + (ab'c) = (b' + c')(b + c + d')$$ using an axiomatic proof (i.e. using only the basic axioms and theorems of Boolean algebra).However, no matter ...
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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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The number of Balanced Boolean functions

Suppose we have n-variable Boolean function (BF) and we know that the weight of a Balanced BF is $2^{n-1}$. The total number of BFs are $2^{2^n}$, Affine BFs are $2^{n+1}$ and Linear BFs are $2^n$. In ...
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Boolean Algebra Simplification

How do I simplify the following equation? $\newcommand{\pn}{\phantom{\neg}}$ $$\begin{align*} \neg A\pn B \neg C \neg D\\ + \pn A\neg B\neg C\neg D\\ + \neg A\neg B\neg C\pn D\\ + \pn A\pn B\neg C\pn ...
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63 views

Intuition behind duality principle?

I'm looking for an intuitive explanation of the duality principle. I found this proof but it was way above my head, considering I just started Boolean Algebra a couple of days ago. I suspect most ...
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How to find a Boolean expression for a combinational logic circuit?

How to find the logic expression for a logic circuit? For example, this one. I am unsure what the circles before the gates exactly mean.
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171 views

Simplifying boolean expression A'(B'C + BD) + A(D(B'C + BC'))

I went from this A'B'CD' + A'B'CD + A'BC'D + A'BCD + AB'CD + ABC'D To this A'(B'C + BD) + A(D(B'C + BC')) Steps: ...
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3answers
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Show that (P→Q) ∧ (Q→R) is equivalent to (P→R) ∧ [(P↔Q) ∨ (R↔Q)]

I literally have no idea how to start this proof. I get to (P→Q) ∧ (Q→R) = (¬P ∨ Q) ∧ (¬Q ∨ R) and then I get stuck.
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3answers
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Boolean Algebra - Why does (x'y' + x'y + xy' + xy) = 1

Have the answers to my Design Fundamentals homework but I do not know how they got the answer they did without $(x'y' + x'y + xy' + xy)$ equaling $1$. Thanks
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1answer
42 views

Simplifying a logic function using boolean algebra

I have the the following logic function (where $'$ is NOT) $f(a, b, c) = abc + ab'c + a'bc + a'b'c + ab'c'$ I have to simplify it as much as possible using only boolean algebra (no truth tables, ...
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Going from (p ∧ ~q) ∨ (~p ∧ q) to (p ∨ q) ∧ (~p ∨~q)

I am confused on how to go from (p ∧ ~q) ∨ (~p ∧ q) to (p ∨ q) ∧ (~p ∨ ~q). I know they are equal because I plugged them into a truth table and all of the rows have the same values. What would be some ...
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Constructing order embeddings between Boolean algebras from embeddings from their finite subalgebras

Suppose that $A$ and $B$ are two complete atomic Boolean algebras and $R$ is a relation between $A$ and $B$ with the following property: If $Rab$ and $A^\prime$ is a finite Boolean subalgebra of ...
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15 views

How to directly translate a boolean function to a boolean formula which expressed by conjunctive normal form?

How to interpret the conjunctive normal form to a practical meaning?
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Where am I going wrong with this Boolean simplification problem?

I am self-studying the Nand2Tetris course. I am trying to simplify the Or logic gate as much as possible to simplify my HDL-specified circuit. Using the Sum of Products, I write the following for ...
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30 views

Proper ideal of Boolean ring

Let M be proper ideal of Boolean ring R. Which of the following is/are true? 1.$R/M$ is Boolean ring. 2.$R/M$ $\cong$ $Z_2$ if and only if M is maximal ideal.
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31 views

How to know the boolean formula of a boolean function?

Suppose A binary boolean function is showed by a true table. How can I know the (simplest) boolean formula which is interpreted by that function?
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63 views

Curious identity involving symmetric difference

While studying the properties of measurable null sets, I found the following identity: $\bigcup_i B_k\triangle B_i=\bigcup_i B_i - \bigcap_i B_i $ Or in other words, the value of the expression is ...
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1answer
38 views

Follow-Up Help with Truth Tables

I've been trying to solve this circuit problem(and understand it frankly), and I wanted to double check my thought process with the community helpfully. After running the circuit out, I have $A+ \bar ...
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Simplify a boolean algebra expression: xy + xz' + x'yz

I need to simplify xy + xz' + x'yz into xz' + yz. I know that these expressions are equal in truth value, but I'm not sure how ...
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K-Map reduction

There's an exercise which states that depending on certain rules a led(of different colour) shall turn on or not. There are four leds, so I've made four functions (One each led, through Karnaugh Map ...
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how solve this boolean algebra F=A⊗B⊙C=

the function is F=A⊗B⊙C I need to apply De Morgan’s Laws and after that reduce the equation to the simplest form off-course I know how to apply De Morgan’s Laws and reduce but I'm confused about ...
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2answers
56 views

How to simplify this Boolean expression

F=(A+B+C)(A+B+C')(A+B'+C') I used sop method and I am left with A+BC', so the above expression should leave me with (A+B)(A+C'). Iam not able to get to this answer. Help is appreciated.
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61 views

Prove $(A\wedge B)\vee(A\wedge-B\wedge C)\vee(B\wedge-C)=(A\wedge C)\vee(B\wedge-C)$

Let A, B and C be digital inputs. Prove that the following boolean equation holds true for any given values for inputs. (A AND B) OR (A AND (NOT B) AND C) OR (B AND (NOT C)) = (A AND C) OR (B AND ...
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The negation of an implication statement

$$\neg(A \longmapsto B)\lor \neg B$$ Does this this expression simplify to:? $$\neg A\longmapsto\neg B\lor \neg B$$ Which further simplifies to: $$\neg A\longrightarrow\neg B$$
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Boolean Algebra and Godel

Can anyone give an example of a theorem in Boolean Algebra that isn't immediately obvious to someone with a computer that can construct a truth table? Clearly no propisition that can be proved using ...
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Simplifying a function using POS and boolean algaebra

I have a function, $$ f = (A+B\cdot \overline C) $$ I am trying to simplify it this form using the inverse function $\overline f$ from the truth table (by anding the rows which form a '0' result). ...
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1answer
32 views

Simplifying from POS using boolean algeabra

I have a boolean function, f expressed in the Product of Sum form. $$f = (A+B+C)\cdot(A+B+ \overline C)\cdot(\overline A + \overline B + \overline C) $$ On simplification I get, $$ f = ((A+B) + (C ...
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63 views

simple exercise in Cylindric algebra

I am trying to gain a better understanding of cylindric algebra, so I made up this example. Given a general rule that someone's father's father is his/her grandfather: $\forall_X ~ \forall_Y ~ ...
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Matrix representation of Boolean algebra?

Is there such a thing as matrix representations of Boolean algebra? Give a boolean algebra with finite elements {a,b,c...} and operations $\cap, \neg$, we can regard $\cap$ as matrix multiplication ...
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Dense subset relation

Defn Let $B$ be a Boolean algebra. A subset $D$ of $B$ is called b-dense if for every $0\neq b\in B$, there is $0\neq d\in D$ such that $d\leq b$. Defn Let $T$ be a topological space. A subset $D$ of ...
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Functions for boolean operators, that return 1 or 0

Are there any purely mathematical expressions that are equivalent to boolean operators and return $1$ or $0$? For example: $a > b$ Is there any $f(a, b)$ for which if $a>b$, $f(a,b)=1$ and if ...
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Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
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XOR with multiply operation.

can I do that $((A*5) \oplus A)==A*(5\oplus1)?$ and that $(A \oplus B/2) == ((2*A) \oplus B)$? Thanks.
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1answer
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Weak Amalgamation Property for Boolean algebras

I'm trying to study universal algebra and lattice theory by myself. Just got stuck with an exercise from Gratzer's "General Lattice Theory" and it seems to me that I don't fully understand the notion ...
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What if I am not given the labels of a Karnaugh map?

Simplify this expression represented by the map $$\begin{matrix} 1 & 1 & 0 & 1\\ 0 & 0 & 0 & 0\\ 1 & 1 & 1 & 1\\ 1 & 1 & 0 & 1 \end{matrix}$$ ...
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2answers
53 views

'Algebraic' way to prove the boolean identity $a + \overline{a}*b = a + b$

For me, it is pretty clear that $a + \overline{a}*b = a + b$, because the first $a$ in the or will make sure that if the second term must be 'evaluated', $a$ will ...
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Finding boolean/logical expressions for truth table + explanation [closed]

I'm having very hard time finding boolean expressions from truth tables. I've also tried many tricks but still can't get through...think you guys can help me with this??...you'll be doing this lil ...
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The empirically-obvious statement about minimization of Boolean functions

The statement: $\forall f,g: \{0;1\}^n \to \{0;1\} \; (n > 0),$ if $$|f^{-1}(1)| > |g^{-1}(1)|$$ then $f$ has the (non-strictly-)simpler minimization than $g$. $\text{ }$ As mentioned, the ...
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Logical operations precedence and calculator program

I write the C library intended to be used in evaluating math expressions. It should support boolean algebra also. So at the moment I'm stuck with boolean precedence. I'm not a mathematician so that's ...
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What to do with a hanging $1$ in a Karnaugh map?

I am learning about Karnaugh maps to simplify boolean algebra expressions. I have this: $$\begin{bmatrix} & bc & b'c & bc' & b'c' \\ a & 0 & 1 & 1 & 0\\ a' & 1 ...
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1answer
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Simplify Boolean equations

I have simplified following Boolean expressions. Can somebody tell me whether they are right or wrong? 1) F1 = ~(~A ~B C + ~(AB)C) ~(~A ~B C) = ~(~A) + ~(~B) + ~C -------> Apply DeMorgan's law to ...
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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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Boolean algebra: $(x+y)(x’+z)(y+z) = (x+y)(x’+z)$

Could someone explain to me how this simplification is derived? $(x+y)(x’+z)(y+z) = (x+y)(x’+z)$
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Boolean Equivalence using Karnaugh Maps

If I had two functions, where each letter represents a state: f(1) = CD + AB f(2) = AC + AD + BC How could I find the minimum term that would need to be added to the second function to make the ...
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Dense Boolean subalgebras

I was reading this page and, in the third part of the first remark I found the definition of dense sub-algebra of a Boolean algebra. It is stated that there are various equivalent definitions of this ...