# Questions tagged [boolean-algebra]

Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Use this tag for questions about Boolean algebras as structures, or about functions defined from/to Boolean algebras. For Boolean logic use the tag propositional-calculus.

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### Any two points in a Stone space can be disconnected by clopen sets

Let $B$ be a Stone space (compact, Hausdorff, and totally disconnected). Then I am basically certain (because of Stone's representation theorem) that if $a, b \in B$ are two distinct points in $B$, ...
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### Boolean Algebra, Simplification: Don't know the method used

Here's the Karnaugh map: The answer I should be getting from the Karnaugh should be: T = R ∙ (CGM)' I'm really not seeing how this was arrived at through any ...
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### Changing conjunction and disjunction in equivalent boolean functions.

I have some difficulties to prove that if two equivalent boolean functions contain only ∧, ∨ and $\neg$ than we can change ∧ to ∨ and vice versa and the result functions will remain equivalent. There'...
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### An “atom” in Boolean algebra

Could someone explain what an atom in Boolean algebra means? I am acquainted with ring theory and group theory but not Boolean algebra. As far as I can tell from browsing around, it is something like ...
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### A free boolean algebra

Consider the following definition: The boolean algebra $A$ is generated freely with the subset $G \subseteq A$ if for every boolean algebra $B$ and map $f:G \mapsto B$ there is precisely one ...
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### Difficulty understanding why $P \implies Q$ is equivalent to P only if Q.

I have difficulties understanding why $P \implies Q$ is equivalent to P only if Q. I do understand that in the statement "P only if Q", it means if $\lnot Q \implies \lnot P$". Regarding this ...
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### Boolean algebra question.

Is there a way to show that $$A\bar{B}C\bar{D}+D=A\bar{B}C+D$$ using the rules of boolean algebra? I tried several methods such as expanding D with $$D(D+\bar{D})$$ or adding $$D\bar{D}$$ to the ...
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### Lindenbaum algebra is a free algebra

The following is a continuation of this question. I would like to prove that the Lindenbaum algebra is a free algebra. Hopefully I would like to hear hints on how to proceed in the 'right' direction....
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### Any two countable atomless Boolean algebras are isomorphic

How to prove that Any two countable atomless Boolean algebras are isomorphic. This is an exercise of Jech - Set theory book for which I have some difficulty.
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### Why do I keep running into contradictions in this problem (Knights and Knaves variation)?

Edit: I've attempted to solve this another way and posted it as a possible answer. Hesitant to accept it, and would appreciate if anyone could go over it and confirm it's the way to go. There is an ...
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### The completion of a Boolean algebra is unique up to isomorphism

Jech defines a completion of a Boolean algebra $B$ to be a complete Boolean algebra $C$ such that $B$ is a dense subalgebra of $C$. I am trying to prove that given two completions $C$ and $D$ of $B$, ...
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### Non-isomorphic atomless Boolean algebras

All countable atomless algebras are isomorphic. Can one give an example of a pair of mutually non-isomorphic atomless Boolean algebras of cardinaliy continuum?
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### Is there a logic gate (NAND, OR, etc.) which forms a group under the set $\{0,1\}?$

Many of the binary logic operators satisfy algebraic properties like associativity, closure, and having an inverse. I was wondering if you could form a group, or any other algebraic structure using ...
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### Counting Rows of a Truth Table that Satisfy a Condition

How can I mathematically count the number of rows in a truth table of n-inputs that will satisfy a certain boolean condition? For example, say I have a 4-input truth table that will in turn have 16 ...
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 $f_1$ ...
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 ...