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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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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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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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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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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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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showing Boolean algebra equality

I have this exercise in my worksheet : Show that x (z ⊕ y) = xz ⊕ xy I reached this in solving it , but didn't reach the final equation x(z'y + zy') xz'y + xzy' please can someone show how
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Boolean algebra operation precedence?

In my discrete mathematics class we wrote down the truth table for some Boolean functions and in that table they go in the following order: ¬, ∧, ∨, →, ~, ⊕, |, ↓ So, I assumed that this is the ...
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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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Can Boolean ring without unit be embedded into a boolean ring?

While going through a book (Lectures on Boolean algebra, Halmos) I got struck at the following question : Prove that every Boolean ring without a unit can be embedded in a Boolean ring with a unit. ...
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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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Solving special boolean equation set

I have boolean equation sets that look like this (where ^ means xor): eq 1: x1^x3^x5^x6^x9^x10^x11^x13^x17^x18 = 0 eq 2: 1^x1^x3^x10^x12^x17 = 0 eq 3: 1^x2^x3^x5^x8^x10^x14^x16 = 0 ...
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169 views

Convert expression to NAND only

Endless youtube videos and reading through notes later I am yet again stuck. I have to covert the following to NAND only $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot ...
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Trying to simplify the expression $A'B'C'D' + A'BC'D' + A'BC'D + AB'C'D$

So far I've got: $A'B'C'D' + A'BC'D' + A'BC'D + AB'C'D$ $= A'C'D'(B' + B) + C'D(A'B + AB')$ $= A'C'D'(1) + C'D(A \;\text{ XOR }\; B)$ $= C'[A'D' + D(A \;\text{ XOR }\; B)]$ Did I do this ...
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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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Is the following sentence a tautology: $(p\Rightarrow q)\vee(r \Rightarrow p)\vee(r\Rightarrow s)\vee(r\Rightarrow q)$?

If both $p$ and $q$ are false then ($p\Rightarrow q$) is true. If either $p$ or $q$ is true then one of ($r\Rightarrow p$) or ($r\Rightarrow q$) is true. If both $p$ and $q$ are true then all are ...
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Simplifying $(A'+ D) \times [(B \times C) + D']$ in Boolean algebra

$(A'+ D) \times [(B \times C) + D']$ I can rewrite as $(A' + D) \times (D' + B) \times (D' + C)$. Is there a way of simplifying to get to the step of $D' + D = 1$ or $D' \times D = 0$ or any other ...
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Not every boolean function is constructed from $\wedge$ (and) and $\vee$ (or)

Prove that not every boolean function is equal to a boolean function constructed by only using $\wedge$ and $\vee$. Here is my solution, can I ask for a feed back on my solution please? $p∧q$ ...
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Boolean function construction [duplicate]

I need some proof on this statement that not every boolean function is equal to a function constructed by only using ∨ and ∧. I need a boolean function that does not constructed using ∧ and ∨ which I ...
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Homework: Conjunctive Normal Form

The way I understand CNF is as an expression containing AND's of OR's. So an AND-GATE with 3 inputs (A, B and C) should just be A AND B AND C. But apparently this is incorrect. Any guidance would be ...
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Boolean functions built from $\wedge$ and $\vee$ [duplicate]

Prove that not every Boolean function is equal to a Boolean function constructed by only $\wedge$ and $\vee$. Please can you help me giving some hint.
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What is a Boolean Function?

Please explain to me what a Boolean function is, and how do I make an expression. If the statement states that $f=$"she is out of work" and $s=$"she is spending more", how can I write symbolically ...
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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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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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truth table for followings

Hi I am new to this site. I got an assessment to complete tomorrow. Its about Computer programming, and i am having trouble with these questions. Can anyone please help me. Using truth tables show ...
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Example of a function between boolean lattices that preserves $(\top,\bot,\wedge,\vee)$ but not complements.

Its easy to find boolean lattices $A$ and $B$ together with a function $f : A \rightarrow B$ such that $f$ preserves both top and bottom elements, as well as binary meets, but not complements. ...
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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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One implication (on Measure)

Please be noted that charges are finitely additive measures and measure are countably additive ones. Theorem 2.1. Let $\mu$ be a charge on a Boolean algebra $B$. Each of the following conditions ...
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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 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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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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Simplification of boolean expressions

Question 1 $$\begin{align*}A'BC + AB'C + ABC + A'B'C' &=A'B C + B' C (A +A') + ABC\\ &=A'B + C + B'C + ABC \end{align*}$$ Question 2 $$\begin{align*}A'B + B'C + CB &=A'B + C(B +B')\\ ...
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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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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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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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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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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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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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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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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 ...