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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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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Boolean operator precedence

I have a confusion with the following proposition $p\leftrightarrow q\leftrightarrow(p\land q)\lor(\lnot p\lor\lnot q)$ how does this work? can it be $p\leftrightarrow(q\leftrightarrow(p\land ...
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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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'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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boolean smplification into canonical

Expand the following Boolean functions into their canonical form: (i) f(X,Y,Z) = XY+YZ+(X ) ̅Z + X ̅ Y ̅ (II) f(X,Y,Z) = XY+ X ̅ Y ̅ +(X ) ̅YZ
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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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1answer
30 views

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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1answer
29 views

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 ...
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Where do I start with $\sim((P\wedge Q)\vee \sim(P\vee Q))$?

can anyone tell me in a table form how to start with this $\sim((P\wedge Q)\vee \sim(P\vee Q))$ I am confused on how to do this part $\sim(P\wedge Q)$, which one we do first, inside brackets or ...
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19 views

Nand this Boolean Algebra Function?

I'm trying to convert this Expression that I got from minterms given to me by my professor to use only NANDS. I swear it should be right, but the output Multisim is giving me is false. ...
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31 views

how to apply laws of boolean algebra to solve boolean expression [closed]

V=(A+B+C) . (A'+B'+C'). A How to simplify above Boolean-Expression,How to apply Boolean laws
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How can I calculate if a given point is wrapped inside a pentagon?

If I have a pentagon and I know the coordinates of it's nodes, how do I calculate if a point is wrapped inside it? An example to clarify what I mean: Assume that we know the coordinates of the ...
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1answer
25 views

Proof of the identity of a Boolean equation $Y+X'Z+XY' = X+Y+Z$

How to prove the following the identity of a Boolean equation? $$ Y+X'Z+XY'=X+Y+Z $$ I have tried : $ \space\space\space\space\space Y+X'Z+XY'\\ =X'Z+XY'+Y\\ =X'Z+XY'+Y(X+X')\\ ...
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58 views

Are there further transformation principles similar to the Inclusion-Exclusion Principle (IEP)?

This question is motivated by the elaboration of the question Combinatorial Proof of Inclusion-Exclusion Principle (IEP). Let's consider the following two aspects: 1.) IEP transforms at least ...
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Simplifying a Sum of Products expression

I'm having some trouble with reducing the Sum of Products expressions for some questions on an upcoming exam. Below is the table (which is correct) for the first part of the question, the second part ...
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Finding the contrapositive of the statement “I go to school if it does not rain”

I got this question in a exam.There were two more statements in the examination(but they were quite clearly wrong).However I got stuck between these two statements.The contrapositive of the the ...
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1answer
33 views

The ability of a logical statement to represent a two-place truth function.

How can i determine which two-place truth functions can be represented using a logical statement built out of a subset of two logical connectors in $ \{\rightarrow, \wedge, \vee ,\equiv \}$ ? for ...
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Boolean Expression Simplification (De Morgan's)

I need to prove that: $$ !(!(X.W) + !(X.Z))) = X.W.Z $$ I have tried multiple approaches but cannot figure this out. Using DeMorgan's theorem, I break the negative sign binding $XW$, and $XZ$, and ...
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15 views

Finding number of Boolean algebras

How many Boolean algebras are there with four elements $0,1,a,b$ ? I don't know how to proceed with this. Any ideas ?
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Simplifying Boolean algebra question

I'm not quite sure how to go about simplifying this boolean expression, any help would be great. X'Y'+X'Z'+Y'Z
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29 views

Converting large terms to disjunctive normal form (logic)

So hello everyone, I am doing some boolean logic and I have this exercise to convert the following term to DNF (disjunctive normal form), but it is so large that everything I try ends up being mega ...
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Simplify Boolean Algebra Expression

The problem is to simplify: $$ xz+\bar{x}y+zy $$ I have an answer key that says the answer is: $$ xz + \bar{x}y $$ I have no idea how they got this expression, though. The first thing I tried was to ...
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Is it possible to express “$P\leftrightarrow Q$” as a formula in $\to,\neg$ with $P$ only appearing once?

I want to write a propositional logic formula for the biconditional that only uses one side of the biconditional once in the formula. I expect it is impossible, but can anyone think of a proof? There ...
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1answer
24 views

Finding a simple function for a given Karnaugh diagram

I came across one question in which the Karnaugh map for some function is given and using it, i have to find a simple function which gets mapped onto that map. My Attempt: Corresponding to every ...
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How to define equivalency as NAND only function?

I am struggling a bit with boolean algebra. I need to represent equivalency as NAND only function. $(A * B) + (-A * -B)$ I am trying with the Morgan rule but I don't know if I can do that: $(A * ...
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Truth-functional completeness

Let the statement $?PQR$ be determined by the following truth-table. ...
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Need help proving that $ fRg \Leftrightarrow fg = f $ on $ B^{n} $ to $ B $ if and only if $ f(b_1,…,b_n) \leq g(b_1,…,b_n) $

I'm trying to gather my thoughts for proving the following claim: For $ fRg \Leftrightarrow fg = f$ on $B^{n}$ to $B$, show that $ fRg $ if and only if for any input values $ b_1,...,b_n $, we ...
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How can I rewrite this xor formula to generate cnf formulas

$$ \bigwedge_{c=1}^n\bigwedge_{i\epsilon S}\bigoplus_{r=1}^nX_{irc} $$ I have tried $$ \bigwedge_{c=1}^n\bigwedge_{i\epsilon S}\bigwedge_{r_1=1,r_2=1}^n(X_{ir_1c}\vee X_{ir_2c})\wedge(\neg ...
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Boolean Logic - Reduction - $a \vee (a \wedge b) = a$

How would I simplify / reduce the following equation using boolean identities/proofs? $$a \vee (a \wedge b) = a$$ So far I've used the distributivity identity and got $$(a\vee a) \wedge (a\vee b)$$ I ...
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Do brackets around negation signify negating the input or output - Boolean Algebra Logic Circuits

I know that $\overline{p + q}$ will result in the input to the logic gate being p, and q, and we can negate this by using an or gate, followed by a not gate, or we can just use a nor gate. However, ...
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Simplify this boolean algebra?

$$ \begin{align} &\lnot x_1(x_2\land\lnot x_3\lor x_3)\lor x_1(\lnot x_2\land\lnot x_3\lor x_2\land x_3)\\ &=\lnot x_1\land x_2\land\lnot x_3\lor\lnot x_1\land x_3\lor x_1\land\lnot ...
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If $A_1\cap…\cap A_n \neq \emptyset$, does $(A_1\cap…\cap A_n)^{c} =A_1^{c} \cup … \cup A_n^{c} = \emptyset$?

If I have some collection of sets such that $A_1\cap...\cap A_n \neq \emptyset$, then what happens if I apply the complement (denoted by superscript c) to both sides? i.e., $(A_1\cap...\cap A_n)^{c} ...
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Boolean algebra. For all x, y, and z in B, if x + y = x + z and x × y = x × z, then y = z.

In the statements below, $B$ is a Boolean algebra with $\times$ and $+$ for binary operations and ($\bar{a}$)is the complement of $a$. 4.) For all $x$, $y$, and $z$ in $B$, if $x + y = x + z$ and $x ...
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For all $a$ and $b$ in $B$, $(a \times b) + a = a$.

In the statements below, $B$ is a boolean algebra with $×$ and $+$ for binary operations. 3.) For all $a$ and $b$ in $B$, $(a ×b) + a = a$. This is what I have as an answer. Can someone confirm ...
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Why is this a boolean algebra

Let $A = \{a,b\}$. The $\mathcal P(A) = \{\emptyset,\{a\},\{b\},A\}$. Let $+$ be $\cup$, $\cdot$ be $\cap$, complement be set complement, $1$ be $A$, and $0$ be $\emptyset$. I need to explain why ...
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Express a + bc +a'bc'd in the NOR form using De Morgans Laws [duplicate]

I need to Express a+bc+a'bc'd in the NOR form using De Morgans Laws, Any help would be much appreciated.
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How can i turn the Boolean Equation pq+r into a switch circuit?

How can I turn the Boolean Equation $pq+r$ into a switch circuit? I have synthesized this and drawn the NOR gates circuit however I'm not sure how to go about drawing/constructing the switch circuit.
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Applying De Morgans Laws to $a+bc+\overline{a}b\overline{c}d$ in terms of the NOR operator

I need to synthesize $f=a+bc+\overline{a}b\overline{c}d$ into the NOR form. Can I split this since I know that $a+bc=(a+b)(a+c)=\overline{\overline{a+b}+\overline{a+c}}$? I'm just not sure how to go ...
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Applying De Morgan's to express $pq+r$ in terms of NOR operator

In Boolean Algebras I have $pq+r$ which I think is the same as $(p+r)(q+r)$. Now, I need to use De Morgan's laws to synthesize this into the NOR form but I am not sure how to apply the laws here.
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Why is chosen for intersection instead of union?

Constructing a commutive monoid having idempotent elements (the underlying monoid of a Boolean ring) free over a set $X$, I arrive on a very natural way at monoid $M$ having the finite subsets of $X$ ...
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What is the number of self dual boolean functions?

The dual of a Boolean function $F(x_1,x_2 \dots x_n,+,\bullet)$, written as $F^D$, is the same expression as that of $F$ with $+$ and $\bullet$ swapped. $F$ is said to be self-dual if $F=F^D$. What is ...