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 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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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 ...
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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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23 views

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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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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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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15 views

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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29 views

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
126 views

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
51 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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2answers
61 views

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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1answer
24 views

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
50 views

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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23 views

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
38 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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1answer
49 views

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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21 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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1answer
37 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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2answers
48 views

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
28 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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63 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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65 views

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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2answers
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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
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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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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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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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1answer
20 views

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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61 views

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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25 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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3answers
26 views

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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42 views

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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2answers
63 views

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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3answers
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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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1answer
42 views

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 ...