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 algebra: DeMorgan's law confusion

the following function should be put into table values: $$y = \overline{(a*b*d+c)}$$ So the first thing i am doing is using DeMorgan to get rid of the "whole-term-negation": $$y = (\tilde a + \tilde b ...
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boolean algebra: simplify $ a* b *d + \tilde a *\tilde c*d + b* \tilde c* d$

Simplify the following function(algebraically): $$y = a*b*d + \tilde a *\tilde c*d + b *\tilde c *d$$ the solution is: $$a*b*d + \tilde a * \tilde c * d$$ which i checked via karnaugh and also ...
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Larger circuit design for same boolean function?

I've designed this circuit with 4 logic gates, and did Karnaugh map's simplification and Quine McCluskey method. However I found out that actually my circuit design is already optimized and I can't ...
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36 views

Distinct Karnaugh Maps grouping?

I got a table truth with some minterns which I mapped to a Karnaugh Map, then I can see an obvious choice for grouping. But I'm wondering wether in this case is possible to do any other different to ...
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232 views

How to write boolean expression as linear equations 2

I just posted How to write boolean expressions as linear equations and asked about a simple example. Here's what we know so far: Suppose a,b,c,d,e ∈ {0,1}. if the boolean expression is: a ≠ b, I ...
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Simplifying a short Boolean expression

\begin{align*} A’B + A’B’C + ABC’C’ + AB’ + AB’C’ &= A’B + A’B’C + ABC’ + AB’ + AB’C’ \\ &= A’(B +B’C) + ABC’ + AB’(C’+1) \\ &= ??? \end{align*} I'm stuck after this. Please help me!!
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How to write boolean expressions as linear equations

I want to convert a set of boolean expressions to linear equations. In some cases, this is easy. For example, suppose $a, b, c$ $\in$ {0,1}. Then if the boolean expression is: $a$ $\ne$ b, I could use ...
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Ideal:Kernel :: Filter:?

I understand that the notion of a filter is in some sense dual to the notion of an ideal, at least in the context of Boolean algebras1. Let $f:{\mathbf A} \to {\mathbf B}$ be a Boolean algebra ...
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Is there a way to prove a boolean operator isn't universal?

In boolean algebra, I could prove an operator is universal by implementing a NAND or NOR gate with it. But is there a way to prove a boolean operator isn't universal? I would like to know a general ...
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205 views

Proving $(xyz)' = x'+y'+z'$

I'm trying to prove that $(xyz)' = x'+y'+z'$ using theorems/axioms. I tried this but I just want to make sure if its the correct route or if I've done something "illegal"/wrong. ...
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137 views

Binomial expansion through combinations.

If you have $(a+b)(c+d)(e+f)$ how can you expand this? Someone was mentioning that you get different combinations so that you get $adf+ade+acf+ace+bdf+bde+bce+bcf$? Is that the full expansion? As an ...
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391 views

Reduce to sum of products

I'm given the equation $F = (x+w)z' + x(y+z) + xz$ The inverse I got is $F' = [(x'+z)*(w'+z)]*[(x'+y')*(x'+z')]*[x'+z']$ To start I would expand $F'= [x'w' + x'z + zw'+zz] * [x'x' + x'z' + y'x' + ...
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Understanding Sum of product and complete sum of product

I have a pair of problems, the first two of my homework, and I'm already unclear on how finding SOP and CSOP for them work. The first: E=xy(1+z)y' It seems like this just reduces to 0, since 1+z ...
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200 views

Infix to Postfix

4 + x / b - a + 5 AND x AND y OR p OR q What is the tree and the postfix of the expression above? I find it tricky because I am not sure if AND has higher precedence than the arithmetic operators, ...
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98 views

Is it properly applied the Quine McCluskey algorithm by this?

I'm writing some code for implementing the Quine McCluskey algorithm and I simply need to clear out if my logic for implementation is ok. I get some number of minterms and combine each of them so ...
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210 views

Proof of $(P\Leftrightarrow Q)\Leftrightarrow((P\Rightarrow Q)\wedge (Q\Rightarrow P))$

Hi I've been working through Applied Mathematics for Database Professionals and I'm stuck trying to proof this equivalence: $$(P\Leftrightarrow Q)\Leftrightarrow((P\Rightarrow Q)\wedge ...
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87 views

Multiply the number $(1001)_{2}$ by 3 digit number

I want to multiply the number $(9)_{10} \rightarrow (1001)_{2}$ by a 3 digit binary number. 1) How I can extract the boolean equations? 2) Make a circuit of it. so what I did is just see what ...
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Can a Karnaugh map be solved in more than one way?

So I understand for doing a simplification by a K Map I should group my 1's (or even 0's) in $2^n$ elements, always trying to grup as much elements as possible. And I can group even adjacent elements ...
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2answers
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How to apply De Morgan's law?

If for De Morgan's Laws $$( xy'+yz')' = (x'+y)(y'+z)$$ Then what if I add more terms to the expression ... $$(ab'+ac+a'c')' = (a'+b)(a'+c')(a+c)?$$
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135 views

Can't simplify this boolean algebra equation

So I've got an expression I have been trying to simplify and have the answer but I can't figure out how to get to it... can anyone help me out? Equation: $(A\wedge \lnot B \wedge \lnot C \wedge ...
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Is it possible to derive all the other boolean functions by taking other primitives different of $NAND$?

I was reading the TECS book (The Elements of Computing Systems), in the book, we start to build the other logical gates with a single primitive logical gate, the $NAND$ gate. With it, we could easily ...
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Logic Circuit Question

1) Write the boolean expression after every GATE 2) Write the boolean expression of GATE 3 3) Try to simplify the boolean expression of GATE3 I need to know if what I did its right + your advice ...
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Boolean simplification, 5 variables

I'm currently learning for my maths exam, and in the part about boolean algebra I came across an exercise that I can't seem to solve. I probably only need the first few steps to get started. $$ (xyz ...
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Simplify Boolean Algebra

How do I simplify the following expression with Boolean Algebra? Please show what you used to simplify so I can understand. $$ABC + AB'C' + ABC' + A'B'C'$$
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Duality principle in boolean algebra

All the definitions I came across so far stated, that if a statement is true, then also its dual statement is true and this dual statement is obtained by changing + ...
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maximal antichain

I don't understand the definition of Jech (set theory) for "maximal antichain". Let $B$ a boolean algebra and $A$ a subalgebra of $B$. $W\subseteq A^+$ is a maximal antichain if $\sum W=1$ and $W$ ...
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Why is $ab + bc + c\bar{a} = ab + c\bar{a}$ true in binary?

I was simplifying the equation of a logic gates problem and I realized that ab + bc + cā and ab + cā followed the same truth ...
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completeness and saturation

Let $B$ a complete Boolean algebra. Suppose, for $\kappa$ cardinal, that $B$ is not $\kappa$-saturated. Then there exists a partition $W$ of $B$. Because of completeness, we have $B=\sum W\in B$. So ...
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218 views

Which law of logical equivalence says $P\Leftrightarrow Q ≡ (P\lor Q) \Rightarrow(P\land Q)$

I'm going through the exercises in the book Discrete Mathematics with Applications. I'm asked to show that two circuits are equivalent by converting them to boolean expressions and using the laws in ...
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112 views

Isomorphism Subalgebra

Given, the unit interval $I$ endowed with the Lebesgue measure $\mu$, and let $A$ be the (Boolean) algebra of Jordan measurable subsets of $X$ with respect to $\mu$, (i.e. those sets that satisfying ...
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Rationale behind truth values

I originaly asked a question on Programmers.SE to know why $0$ was consider $\text{false}$ and all the other [integral] values were considered $\text{true}$. That was a huge debate and many said it ...
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Generalization of Boolean OR?

I have been looking at the Boolean OR function and Im trying to find its integral analogue. What I mean is: Boolean AND (x, y) where x and y are Boolean Values with 0 = False, 1 = True is equivalent ...
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Reducing Boolean expressions

Just learning mathematical proof writing and came upon this interesting question Writing an expression using logic. $$(P \land Q \land \lnot R) \lor (P \land \lnot Q \land \lnot R) \lor (\lnot P ...
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A matrix w/integer eigenvalues and trigonometric identity

Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated: Let $n$ be a natural number. (a) Consider the following Toeplitz/circulant symmetric matrix: ...
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Inferring simplest method to convert bit array 1 to bit array 2.

Consider the set of all bit arrays of length $n$. Now consider the set of all 1-to-1 functions that map from this set to this set. Now select a single function out of the latter set. Is there any ...
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Boolean Algebra Transform

I am revisiting Boolean algebra after a long while. Can somebody help show me how to simplify the LHS to get the RHS? $$abc * a'bc + (abc)' * (a'bc)'\quad = \quad \;b'+c'$$
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Filters of Boolean algebras which are Boolean algebras

Looking at some filters generated by elements of a finite Boolean algebra I have the impression that many/most/all of them are by themselves Boolean algebras (at least I didn't stumble upon a ...
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How does it evaluate A XOR B XOR C?

I am trying to solve the following combination, ...
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48 views

How do we go about factorizing boolean expressions?

How do we know how to go about factorizing a boolean expression when there are so many ways? For example, the factorized form of $ABC + A'B'C'$ is $(A + C')(B' + C)(A' + B)$, but how do we know how ...
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Boolean Equation Transformation

Can someone show me the steps in getting from $f = (ab + c')(d' + e + f')$ to $f = abe +ab(df)' + c'e + c'(df)'$? I am trying to relearn Boolean algebra after a long hiatus.
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Prove that $(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$.

Logically, the following proposition makes sense: $(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$ Or, in english, if sets $S$ and $T$ share no elements, and the union of ...
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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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112 views

Measure on Boolean algebra

my question is: Suppose that $\mathfrak{B}$ is a measurable Boolean algebra, does this mean that "Every measure on $\mathfrak{B}$ should be strictly positive ? or this will be the case after ...
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59 views

Simplify Expression Question

Anyone can tell me if I can simplify this expression more? I Simplified this function => $minterm(1,3,4,6,7,9,10,11,12,15)$ to this expression: $W'X'Z+W'Z'X+WYZ+W'XYZ+WX'Y'Z+WX'YZ'+WXY'Z'$ Thanks!
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Single variable true or false statements

If I have a true or false statement S, depending on a varible x, is there some standard function or opperation in formal logic, that takes the statement S and the variable x, and outputs $1$ if ...
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Expressions Simplifications Boolean Algebra

Expressions Simplifications Boolean Algebra I started simplifying function and got to the detailed picture and wanted to know if I can reduce the above expressions, for example : Y'.X'.Y = 0 ? ...
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Boolean Algebra Simplification Question - Proof of equation

Boolean Algebra Simplification Question - Proof of equation I`m trying to proof this equation: X'.Y' + Y'.Z + X.Z + X.Y + Y.Z' = X'.Y'+X.Z+Y.Z' What your are suggesting? to add some ...
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Basis of a Boolean Algebra

I have a construct that I proved forms a (finite) Boolean Algebra of sets over a given universe. My questions are as follows: Do I immediately know that there exists a unique basis for it? If yes, ...
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Simplifying boolean expression: $!(x!z+y!z+xy+z)$

This is the expression: ', ! not+ or $((x'y'+z)'+z+xy+wz)'$ After some steps I can get $!(x!z+y!z+xy+z)$ How can I continue from here?
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Stone's Representation Theorem and The Compactness Theorem

If you're working on $\mathsf {ZF}$ and you assume the compactness theorem for propositional logic, then you have the prime ideal theorem, and thus you can show that the dual of the category of ...