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 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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25 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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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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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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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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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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280 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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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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33 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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41 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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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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58 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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234 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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18 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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48 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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33 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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497 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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32 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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31 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
22 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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21 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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23 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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160 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
26 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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32 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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77 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
26 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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37 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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159 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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70 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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329 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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64 views

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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54 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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30 views

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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1answer
64 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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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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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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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
39 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
352 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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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
62 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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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 ...