# 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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### How many n-ary Boolean functions essentially dependent on each of their arguments?

How many n-ary Boolean functions essentially dependent on each of their arguments? essentially dependent means that $$f(b_1,…,b_{i−1},0,b_{i+1},…,b_n) \neq f(b_1,…,b_{i−1},1,b_{i+1},…,b_n)$$
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### How to convert a mod 2 function to an expression in Boolean Algebra

I'm not sure if this is the right place to post it but I have a question I'm having a hard time understanding. The questions is: Convert the function $X^3Y + 2XZ + WX + W$ mod $2$ to an expression ...
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### Proof for $∃xA⇔¬∀x¬A$

I want to prove, that $∃xA⇔¬∀x¬A$, using classic axioms. I think, I have to start with the following step: $∃xA⇔∃x¬¬A$ But I do not know, how to make this step, using axioms: $∃x¬¬A⇔¬∀x¬A$
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### How to show that if $\neg b = a \land d$ then $a \land \neg b = \neg b$ and $b \land \neg a = \neg a$

I'm new to boolean algebra and am having trouble proving the following simple theorem. Many thanks for any help. If $\neg b = a \land d$ then $a \land \neg b = \neg b$ and $b \land \neg a = \neg a$. ...
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### Simplifying a Boolean Expression 2

The boolean expression is as follows: (¬A^¬B^¬C)∨(A^¬B^C)∨(A^B^¬C)∨(A^B^C) I have found that A⊕(¬B^¬C) is equal to the above but I have absolutely no idea on how to get this result, I have spent ...
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### Inequality with respect to transitivity

Given a relation R, R is said to be transitive if aRb ∧ bRc, then aRc. The unequal relation (≠) is not transitive, for instance a≠b ∧ b≠c, then a≠c is an invalid consequent of the antecedent (a≠b ∧ b≠...
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### Joins in lattices and sublattices

Let $A$ be a lattice, and $B$ be a sublattice of $A$. Why is the join of $A$ included in the join of $B$? That is, why is $\bigcup_{t\in T}^{A} a_t\leq\bigcup_{t\in T}^{B} a_t$? (I am tempted to ...
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### Product of maxterms

Please help me break the ice in understanding how we derive a product of maxterms, say, for: $xy+x'z$ I could be missing some concept here in this but be patient with me. I have also done SOP and ...
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### Is there any way to simplify the following boolean expression?

I was trying to manipulate with litarals and minterms of this booleans expression but it really did not lead to anything that could simplify the expression further.. Not sure if I am doing it wrong or ...
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### Boolean algebra-dual of an expression

Can anyone think of an expression that is equal to its dual ? I've been trying to solve this for the past 2 hours, but nothing comes to mind.
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### Proof for $\forall x A \Leftrightarrow \neg \exists x \neg A$

I try to proof, that $\forall x A(x) \Leftrightarrow \neg \exists x \neg A(x)$ I know how to prove, that $\forall x A(x) \Rightarrow \exists xA(x)$, but I don't understand how to get negation.
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### Implementing logic functions using only an OR gate with one input inverted

I've been looking at logic gates, boolean expressions and Karnaugh maps. I ran into a question regarding whether it was possible to implement all logic functions using only one logic gate: an OR gate ...
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### Why can't a Venn diagram constitute a proof?

I'm reading Boolean Algebra and Its Applications and come across this statement about Venn diagrams: It should be remembered that such diagrams do not constitute proofs, but rather represent ...
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### Boolean Simplification of a large problem

I am unsure where to even start on this problem. My intuition that what ever can be done to the original problem can be done over and over to simplify the whole thing. Please help with some guidance. ...
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Below is my simplification, but my truth tables don't line up, but I can't find my error. $(a+b) \cdot (a \cdot c + a \cdot \overline{c}) + a \cdot b + b$ $(a+b) \cdot a \cdot (c + \overline{c}) +... 2answers 54 views ### Boolean Simplification of$(\overline{a+b+c})+a\cdot(b+ \overline{c})$I'm lost, when checking my answer via truth tables, my simplified form does not match the original equation. My work, with reasoning step by step is below. Can you help me figure out where I'm wrong, ... 3answers 136 views ### What is the most simplified form of$y(x′z + xz′) + x(yz + yz′)\$

I am stuck on a problem that I know the logical answer to, yet I cannot seem to simplify properly to get there. I want to simplify $$F(x,y,z)=y(x′z + xz′) + x(yz + yz′)$$ I know the simplest form (...
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### Boolean and equivalent to summation

Is there a mathematic symbol to express the application of AND operator to a set of booleans, that returns true only of all booleans in the set are true. Something like the summation operator on a set ...
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### If ¬ has a higher precedence than ∨, could one affirm “¬ (p ∨ r) ∨ r” <=> “¬ ((p ∨ r) ∨ r)”?

I'm currently in a disagreement with a colleague over how one should intrepret the precedence of the ¬ operator in boolean algebra, and I hope someone here may enlighten me. We both agree that the ¬ ...
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### How should I think when implementing Patrick's method?

I have implemented Quine-McCluskey method of boolean function simplification. I ended up with the table of prime implicants: As you can see my results are the same as these on wikipedia. However ...