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 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
23 views

How to simplify the following expressions to a minimum SOP form using the basic identities of Boolean algebra? [on hold]

How to simplify the following expressions to a minimum SOP form using the basic identities of Boolean algebra? $$A'C'+C'D'+AC'+A'CD'+A'B'D+A'BD$$ Thanks
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1answer
20 views

Implement using only XOR and AND gates [on hold]

How can I implement the function: F = ABC’D + AD’ + A’D using only XOR and AND gates ?
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21 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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2answers
28 views

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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3answers
102 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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2answers
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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
17 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}) ...
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34 views

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

I'm lost, when checking my answer via truth tables, my simplified form does not match the original equation. My work, with reasoning step by step is below. Can you help me figure out where I'm wrong, ...
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3answers
26 views

What is the most simplified form of $y(x′z + xz′) + x(yz + yz′)$

I am stuck on a problem that I know the logical answer to, yet I cannot seem to simplify properly to get there. I want to simplify $$F(x,y,z)=y(x′z + xz′) + x(yz + yz′)$$ I know the simplest form ...
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1answer
19 views

Boolean and equivalent to summation

Is there a mathematic symbol to express the application of AND operator to a set of booleans, that returns true only of all booleans in the set are true. Something like the summation operator on a set ...
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1answer
23 views

If ¬ has a higher precedence than ∨, could one affirm “¬ (p ∨ r) ∨ r” <=> “¬ ((p ∨ r) ∨ r)”?

I'm currently in a disagreement with a colleague over how one should intrepret the precedence of the ¬ operator in boolean algebra, and I hope someone here may enlighten me. We both agree that the ¬ ...
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0answers
10 views

How should I think when implementing Patrick's method?

I have implemented Quine-McCluskey method of boolean function simplification. I ended up with the table of prime implicants: As you can see my results are the same as these on wikipedia. However ...
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3answers
39 views

XOR of two numbers AND third number ?

If: x & (a ^ b) != 0 Then one of the following holds: x & a == 0; x & b != 0 or x & b == 0; x & a != 0 What is the reason for this? And are there similar ...
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0answers
27 views

How do I know that min-term can't be combined any further?

I'm trying to learn (and implement) Quine-McCluskey algorithm for boolean function minimalisation. I'm learning the algorithm from wikipedia example. From that I understood the following: Take all ...
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1answer
69 views

The ring of idempotents

Let $R$ be a commutative ring. Then its ring of idempotents $I(R)$ consists of the idempotent elements of $R$, with the same multiplication as in $R$, but with the new addition $x \oplus y := ...
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1answer
44 views

Lattice Properties Challenge [closed]

i see in one note that the following is true and false, anyone could help me why? if + be a minimum upper bound and * be a maximum lower bound why this properties be true for Lattice? $ a + (b*c) ...
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1answer
50 views

Find all the prime implicants for the following Boolean functions, and determine which are essential.

Find all the prime implicants for the following Boolean functions, and determine which are essential: F(W,X,Y,Z) = Im(0,2,5,7,8,10,12,13,14,15) Book solution: Prime = XZ, WX, X'Z', WZ' Essential = ...
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21 views

How do you simplify this boolean expression?

Original expression : $$J = ((A′ + B)′ + C′)′ + DC′ + AB′ $$ Here is what I did but the correct answer is $AB' + C + D$. $$J = ((AB') + C')' + DC' + AB'\\ J = ((AB')'C) + DC' + AB' \\ J = ((A' + ...
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3answers
62 views

Difficulty understanding why $ P \implies Q$ is equivalent to P only if Q.

I have difficulties understanding why $ P \implies Q$ is equivalent to P only if Q. I do understand that in the statement "P only if Q", it means if $ \lnot Q \implies \lnot P$". Regarding this ...
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1answer
22 views

Simplify this Boolean expression?

X Y Z+X Y' Z'+X' Y' Z+X' Y Z' I know it simplify to (X XOR Y XOR Z),BUT I want to simplified using only AND, OR, and NOT Gates? Please help I spent three hours but I don't get the same truth table.
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1answer
18 views

Finding Product-of-Maxterms Form

I need help to resolve this problem, i have the following boolean function: [(A.!C)+!(A.!C)].!(A.!B) The Truth table is: (please see this LINK TO wolframalpha ...
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2answers
32 views

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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5 views

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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2answers
44 views

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
22 views

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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0answers
29 views

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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0answers
8 views

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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7 views

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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43 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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21 views

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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1answer
49 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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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
18 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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34 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, ...
2
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2answers
23 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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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
35 views

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
20 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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1answer
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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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
24 views

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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56 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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60 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: ...