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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Explanation of Paragraph on Measure Theory

I have seen this situation in some other paper, it says: The properties of $m$ can be transferred to $\mu$. I am asking can this be the case for example: If $m$ is non-atomic, then $\mu$ is also ...
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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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Writing a boolean formula and logic circuit that computes mux

Let $mux(p_{11}, p_{10}, p_{01}, p_{00}, x_1, x_0) = P_{x1x0}$ (with all variables bits). Write a boolean formula, and then draw a circuit, that computes mux. For ...
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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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self dual boolean function

How many self-dual Boolean functions of n variables are there?Please help me how to calculate such like problems. A Boolean function $f_1^D$ is said to be the dual of another Boolean function ...
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One implication (on Measure)

Please be noted that charges are finitely additive measures and measure are countably additive ones. ![enter image description here][1] ![enter image description here][2] for charges $ ...
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Do these equivalence classes have any special property!

Suppose that $\lambda$ be a measure on the interval $I=[0,1]$, and Let $\mathcal{N}$ be the family of null sets. It is known that "measure algebra $\mathcal{B}$ " is the algebra of all measurable ...
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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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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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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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calculating number of boolean functions

I would just like to clarify if I am on the right track or not; I have these questions: Consider the Boolean functions f(x,y,z) in three variables such that the table of values of f contains exactly ...
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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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Examples of boolean functions in conjunctive normal form

Give an $n$-variable Boolean function $f(x_1; x_2; \cdots ; x_n)$ in conjunctive normal form so that $f$ is $1$, respectively, (a) If at least one of the $n$ variables is $1$; (b) If at most one of ...
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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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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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314 views

Logic Circuits And Equations Issue - Multiply Binary Number By 3

I am trying to build a logic circuit that multiplies any 4 digit binary number by 3. I know that if I multiply\divide number by 2 it moves left\right the digits, but what I`m doing with multiply by ...
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443 views

problem simplifying boolean algebra expression using consensus theorem

Please simplify this logic expression for me with helping boolean algebra : A'C'D + A'BD + BCD + ABC + ACD' I know that must use consensus theorem . my solve : STEP 1 : Terms 1 & 3 ...
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Is this a good enough proof?

Is this proof good enough? If not, any feedback would be appreciated. Thanks. Either exhibit $333 $ different boolean functions on the three variables $p; q; r,$ or prove that there aren’t $333$ ...
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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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59 views

Checking Boolean Algebra work - Simplification

I am currently working on an assignment for a CE class I am taking, and I wanted to know if I have been simplifying these equations correctly. I'm supposed to reduce them to a sum of products. 1) ...
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398 views

Simplify the following expression using Boolean Algebra into sum-of-products (SOP) expressions

Simplify the following expression using Boolean Algebra into sum-of-products (SOP) expressions $Q.S.U + (Q' + S').(R + V) + U.(R + V) + Q' + S.T.U$ $.$ = AND $+$ = OR This is what I have so far ...
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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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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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48 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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Boolean Algebra: Simplifying multiple XOR and XNOR

Is there any way to simplify a combination of XOR and XNOR gates in the following expression? I have tried multiple boolean theorems and I have not been able to simplify this any further: The ...
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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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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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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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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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47 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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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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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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Trying to simplify $A'B'C'D + A'B'CD + A'BC'D + AB'CD + ABCD$

My solution so far: $A'(B'C'D + B'CD + BC'D) + A(B'CD + BCD)$ $= A'(C'D(B' + B) + B'CD) + A(CD(B'+B))$ $= A'(C'D(1) + B'CD) + A(CD(1))$ $= A'C'D + A'B'CD + ACD$ $= D(A'C' + AC) + A'B'CD$ $= ...
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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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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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56 views

Quotient of boolean algebra by an ideal

The homomorphism theorem states that every boolean ideal $I$ of a boolean algebra $A$ is the kernel of a boolean isomorphism. I'm reading a paper where the author presents a short proof of this ...
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How do Boolean-valued functions work?

Consider this function: $$P: X\to \{true, false\}.$$ There's nothing in that expression that says when $X$ is true and when it is not true. How do these work?
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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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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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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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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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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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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 ...