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 Simplification

I have to simplify A'BC' + A'B'C + A'BC + ABC My result was A'BC Is this correct?
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Need to create a specific four variable Boolean function?

I need to create a Boolean function consisting of four inputs (a,b,c,d) that is 1 if no more than 2 of its inputs are 1. I have experienced great difficulty on this particular problem. If anyone ...
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Simplification for the boolean expression AB'+A'B [on hold]

How to simplify the boolean expression AB'+A'B
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Please simplify this expression [on hold]

Can someone help me with this expression and how to simply it. Simply this expression : A'B'C + A'BC + AB'C' + AB'C + ABC
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Boolean algebra simplification of ( (A'+BC)' + (AB')')' [on hold]

Can anyone help me how to simplify this expression?
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2answers
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Simplification of boolean expression with xor

I need to simplify the following boolean expression ¬(A xor B) xor (B + ¬C) I know A xor B = ¬AB + A¬B Then the expression will become ¬(¬AB + A¬B) xor (B + ¬C) However, I stuck on it and I don't ...
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1answer
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Boolean algebra simplification a'bc+ab'c+abc'+abc

Can't figure out how to simplify $(^\neg a)bc+a(^\neg b)c+ab(^\neg c)+abc$, I'm really bad at this...
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Simplify Boolean Function F(w,x,y,z) = wx' + y'z' + w'yz' using Karnaugh map?

Given: F(w,x,y,z) = wx' + y'z' + w'yz' How would I simplify it? Also given the first term wx' would that translate to 10 or would I have to incorporate y and z ...
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Condition on Vector Boolean Function to be Bijective

Suppose the vector boolean function be $$\begin{align} f:F^n_2 \longrightarrow F_2^n \\ (x_1,\dots ,x_n) \longrightarrow (x_2,\dots x_n,g) \\ \\ g:F^n_2 \longrightarrow F_2 \\ (x_1,\dots ,x_n) ...
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Relatioship b/w Bent function and Correlation Immunity

I assume that the readers know the definition of Walsh-Hadamart transform for Boolean function. An n-variable Boolean function $f$ is called bent if $W_f(\alpha)=\pm 2^{n/2}$ $\quad \forall \alpha ...
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40 views

Equality of two expressions describing a filter

Let $U$, $W$ be boolean lattices with order $\sqsupseteq$, and $U \supseteq W$. The top element of $U$ is the same as the top element of $W$. The bottom element of $U$ is the same as the bottom ...
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How to simplify a boolean expression with only the algebra laws?

The expression is $$AD^{'}+A^{'}B+C^{'}D+B^{'}C $$ It clearly has some form of symmetry because there are exactly an equal amount of complements for each variable. I have to simplify it to a product ...
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43 views

Verify Demorgan's Law Algebraically

If $\overline X \equiv \text { not }X$, De Morgan's Laws are stated as: $ \overline{(A + B)}= \overline A\cdot \overline B$ $ \overline{(A\cdot B)} = \overline A + \overline B$ Verify the above ...
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37 views

Not sure how a boolean algebra simplification step is done?

I've simplified a rather long boolean expression down to (where $'$ is $NOT$, $+$ is $OR$, and multiplication is $AND$): $f = b'd'+ad'+ac'+ab+bc'd$ But the simplest expression that I'm supposed to ...
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1answer
28 views

Demonstrate that (p → q) → ((q → r) → (p → r)) is a tautology.

I'm struggling to demonstrate that (p → q) → ((q → r) → (p → r)) is a tautology. I know that : ...
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1answer
28 views

Demonstrate that p ↔ (p ↔ q) ⇔ q

I know the answer is : (p ↔ p) ↔ q ⇔ q 1 ↔ q ⇔ q q ⇔ q But I don't understand why it isn't : ...
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1answer
12 views

Boolean algebra proof (a+b) (a+c)' = a'bc'

I have to prove that (a+b) (a+c)' = a'bc' My algebra skills are really rusty and I was wondering what identities are used to solve this so I can get a better understanding
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1answer
17 views

Boolean Algebra - What is the meaning of f(x) = ∑(N)?

Suppose you had: f(x,y,z) = ∑(2,3,4,5) What does this represent or map? For each of the variables x,y,z?
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1answer
15 views

Equivalence of the two boolean expression

This is a question from a textbook on digital logic which I am having a difficult time with: Prove that the following expression is valid: ...
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27 views

Convert from sum of products to product of sums (Boolean algebra)

I had to simplify a boolean expression with a k-map then put it into a NOR-gate implementation circuit. I haven't made the circuit yet, but here is the work I've done: Original function: $$F(w, x, ...
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2answers
21 views

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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19 views

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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1answer
21 views

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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2answers
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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
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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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1answer
31 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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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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2answers
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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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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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2answers
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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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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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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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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
29 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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1answer
59 views

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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1answer
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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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1answer
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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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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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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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3answers
53 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
36 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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1answer
44 views

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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1answer
18 views

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
49 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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1answer
56 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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1answer
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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). ...