# Tagged Questions

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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### (P and(not(not P or Q))) or( P and Q) equals P

I've been trying to verify the condition above but I get stuck on the passage : $$(P \land (P \land \lnot Q)) \lor (P \land Q)$$ I don't know how to simplify it since there are two ands and a not Q. ...
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### How to prove these two sets are identical?

This is more a question of the methadology one should use to solve these type of questions: Say there is a set $V \subseteq X \subseteq Y$ and $U \subseteq Y$ such that $$X \setminus V = U \cap X$$ ...
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### Hypercontractivity Lemma

In the proof of the Hypercontractivity Lemma here http://www.cs.cmu.edu/~odonnell/boolean-analysis/lecture13.pdf (3.4) what does it mean to split $p$ into $r + x_n*s$, why can we do this?
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### Complete atomic boolean algebras as coalgebras of some endofunctor on Set

I was hoping to use the fact that CABAs are powersets with extra structure on the morphisms to find an endofunctor $F:\text{Set}\to\text{Set}$ with $\text{Set}^{op}\simeq\text{Coalg}F$. I started by ...
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### Brackets in Boolean ALgebra Distributive Law

What is the purpose of the brackets in all the examples I've seen of the distributive law? Why are there no brackets when distributing an AND term and there are when distributing an OR term? Could I ...
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### AB'+B'C' NOR only gates

a few days ago I had my midterm exams in Boolean algebra, and one question bugs me. The final answer of the question was AB'+B'C' (A and not B or not B and not C), and we were supposed to draw a ...
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### Boolean Algebra x+y=0 proof

So I am having a problem solving this proof of Boolean algebra. I am trying to prove that if x + y = 0 then x = 0 This is what I have tried ...
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### Solving a boolean expression

I am trying to solve the following Boolean expression: $$a + \neg{a} b + \neg{a} \neg b c + \neg a \neg b \neg c d + \dots$$ The question asked was to use Boolean algebra in order to solve the above ...
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### Can anybody explain me how to solve logical equations with using matrices?

Task:I need to find N, with which number of solutions is 32. \begin{cases} (X_1 \land X_2) \oplus (X_1 \land X_3) \oplus (X_2 \land X_3) =X_1 \land (\lnot( X_2 \land X_3)) \\ (X_2 \land X_3) \oplus (...
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### On a theorem concerning free Boolean algebras

In Sikorski's book "Boolean Algebras" (3rd edition), p. 42, one finds the following theorem: In order that $\mathfrak{A}$ be a free Boolean algebra with $n$ free generators, it is necessary and ...
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### Is it possible to solve such system of Boolean equations?

During a discrete mathematics test I got this question. I did not see it in my course and I am baffled, because I don't even know from where to start. All I found on google for such subject were ...
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### Finding the principal disjunctive normal form (PDNF) of a Boolean expression

Find the principal disjunctive normal form (PDNF) of a Boolean expression $$((p\wedge q) \rightarrow r)\vee((p\wedge q)\rightarrow \neg r).$$ I tried by expanding it but I am stuck with the ...
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### Boolean Algebra Product of Sums

I have a question to solve the following expression and get it in terms of product of sums (AB' + A'B)C And I tried taking the compliment of this ...
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### Design a circuit for a light fixture

Design a circuit for a light fixture controlled by four switches, where flipping one of the switches turns the light on when it is off and turns it off when it is on and please explain your answer
I need to prove that given $$f_1 = c + a'd' + bd' \quad\text{and}\quad f_2 = a'b'd' + a'bd' + ab'c + abd'$$ that $f_1 = f_2$. How do I manipulate $f_2$ to be exactly like $f_1$? I have tried a lot ...