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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What are three possible ways to express the following Boolean function with eight or fewer literals?

F= A'BC'D + AB'CD + A'B'C' + ACD' I assumed that the question was asking for me to simplify. I placed the terms into a kmap and have gotten SOP F= A'B'C' + A'C'D + AB'C + ACD' or POS F= ...
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Finding the atoms of a Boolean Algebra

I have a homework question that asks me to find the atoms of the Boolean Algebra that contains 256 Boolean functions "such as F1(x,y,z) = x + y +z, F2(x,y,z) = x + xz, F3(x,y,z) = xyz+ xyz and so on". ...
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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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Logic subject-reductio ad absurdum

Can you solve this using method reductio ad absurdum? 1)A ↔ (¬ B v C) ¬ A ¬ B 2)¬(R∧ (S v T)) 3)R∧¬ T S ¬R∧ S
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Explain why the description defines a Boolean Algebra

This is the exercise: Let $A = \{a,b\}$ and list the four elements of the power set $\mathcal P(A)$. We consider the operations $+$ to be $\cup$, $\cdot$ to be $\cap$, and complement to be set ...
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Discrete Mathematics (boolean)

Either exhibit 333 different boolean functions on the three variables p; q; r, or prove that there aren’t 333 different such functions $p$ $q$ $r$ $0 0 0$ $001$ $010$ ...
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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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250 views

Conjunctive Normal Form vs Product of Sums

I am confused as to what the difference between Conjunctive Normal Form and Product of Sums is. Can someone explain what is different about them? It seems like they both only use groups of OR ...
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Whats wrong with this reasoning…

Suppose I have two non-distinguishable balls (for example two white ones) and I color them with red and green, then a combinatorial reasoning could go like this. Suppose I enumerate the balls, ball ...
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238 views

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 algebra simplification hw

I'm given the equation $F=(a+b+c)(a'+b')(a+b'+c)$ and it's supposed to simplify into a sum of two product terms, each with two literals. I know the answer is $ab'+a'c$, but I'm unsure how to get ...
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How to construct the truth table for a combinational circuit

I am trying to construct the truth table for a combinational circuit with the following conditions : A) Room with 4 doors , 1 light, a switch near each door that controls the light (4 in total) B) ...
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57 views

Simplifying Boolean Algebra Expression with 3 variables

Can someone help me simplify this in Boolean algebra? It should be one step at a time so I can understand it. The expression is: $(x+y+z)(x+z)(x'+y+z)$ I tried doing this: (it's probably wrong, ...
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A boolean algebra is complete if its stone space is extremally disconnected

I have the following proof, but I don't understand one of the steps: Theorem 4.4. A Boolean algebra is complete iff its Stone space is exlremally disconnected. Proof. Identify the given ...
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84 views

What does quotienting by a congruence mean?

I have come across quotient algebras in my different mathematics courses. I know of quotienting with normal groups, quotienting with ideals etc. While studying Boolean Algebra I encounter quotienting ...
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Discrete Mathmematics

Are the boolean functions $(p\wedge \neg q)\vee (\neg r\wedge q)$ and $(p\vee \neg q)\wedge (r \vee \neg q)$ equal? Explain your answer. Here my solution, Please give me a feed back on this solution, ...
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42 views

Homomorphism between a ring which is a boolean algebra and one which is not.

I remember reading in a textbook that there can exist a homomorphism between a ring which is a boolean algebra and one which is not. Can anyone give me some example of this.
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Sigma Algebra: Etymology

Why do we talk of sigma algebras in measure theory. As far as I know sigma is related to the countability. But what does it stand for?
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Boolean Algebra, removal of redundant term

How do I simplify this boolean expression A¬B + A¬C + BC¬D + A¬D to A¬B + A¬C + BC¬D with boolean algebra? The A¬D is redundant, I can see why it is when I examine the truth table for this ...
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Boolean Functions-Algebraic rules for Boolean functions-Associative Rule

Is the function $(p \wedge q) \vee r$ equal to the function $p \wedge (q \vee r)$? Let $a(p,q,r)=(p \wedge q) \vee r$ $b(p,q,r)= p \wedge (q \vee r)$ By associate law $a=b$, but using $a(0,0,1)=1$ and ...
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Expression conversion using de Morgan's laws

I'm sorry strongly, because it's a very dummy question... I have an example in the algebra of logic. I need to convert an expression using the rules of de Morgan - replace by the conjunction of ...
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Boolean Queries in First Order Logic

I understand first order logic and how its constructed but I have some trouble understanding how the following statement and its FO query are formed. This is from a book. ...
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Proving the Boolean expressions

Are these two Boolean expressions the same? *$co$ is the carry out while $ci$ is the carry in.
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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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Can this be simplified any further? (Boolean algebra)

I've been working on this expression, but all my attempts have failed to simplify it further. $$A'.B' + A'.B.C' + A'.B.C + A.B'.C'$$ I have tried to pick out $A'$ based on the distribution law: ...
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boolean expressions simplification Help needed.

I am stuck simplifying. Can anyone help? It states that $$ (XY’+YZ)’ = X’Y’ + X’Z’+YZ’ $$ I tried all axioms yet I can't figure it out.
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Prove $ab + ab\overline{c} + bcd = b(a+c)(a+d)$

Do I need to use absorbtion law to prove them? $ab + ab\overline{c} + bcd = b(a+c)(a+d)$ $ab + cd = (a+c)(a+d)(b+c)(b+d)$. For 1), I simplified $ab+ ab\overline{c} + bcd$ into $b(a\overline{c} + ...
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the quotient boolean algebra of $P(\kappa)$ over the nonstationary ideal

Let $\kappa$ be a regular cardinal. Then the quotient boolean algebra over the nonstationary ideal, $P(\kappa)/I_{NS}$ is $\kappa^+$-complete. Specifically, any $S \subseteq P(\kappa)/I_{NS}$ of ...
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153 views

Two question on ternary Cantor set & Jordan content

Is it true that all subsets of the Cantor set have Jordan content zero? What is the definition of countably generated Boolean algebra? Does the Boolean algebra of subsets $[0,1]$ which ...
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Boolean simplification expression

In words my problem is NOT(p AND q) AND (NOT p OR q) AND (p OR p). I have rewritten it in symbols ¬(p ∧ q) ∧ (¬p ∨ q) ∧ (p ∨ p) I got this far: (¬p ∨ ¬q) ∧(¬p ∨ q) ∧ p Any help please?
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Is my answer for this truth table & boolean expression correct?

I was given the following boolean diagram: I had to come out with the truth table and the simplified expression. I need help to check if my answers are correct below.
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Converting to complements

Let's say I wanted to convert the ands and positive variables to their complements and ors. Would this be correct? $$DE=$$ $$(DE)''=$$ $$(D'+E')'$$ Or another example: $$D'E=$$ $$(D'E)''$$ Can ...
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Back-and-Forth Argument vs. “One-Way” Argument

The wikipedia article on the Back and Forth Argument claims at the end: If we iterated only step $(1)$, rather than going back and forth, then in some cases the resulting function from A to B ...
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61 views

boolean algebra simplyfing

I need to solve these expressions with boolean algebra: $$Z = a'bc+ab'c+abc'+abc\mbox{ AND }Z = a'(b+db'c')+a'b'(d'c'+c)$$ Every advice is more then welcome. Thanks
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Can I factor out or statements on the other side of an equation in boolean?

I have this boolean equation: X'Y'+XY+X'Y=X'+Y I want to prove it. Now I was wondering if I can rearrange this equation, if I could, so I can factor out the ...
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Simplify a Boolean Expression

I have to simplify this w′x′y′z + wx'yz' + w'xyz' I keep getting different answers depending on whether I start on the left or the right of the expression Any advice or help would be appreciated. ...
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A problem with a boolean function.

This is one of my assignment problem and I almost finished others. This question confused me for quite a long time and now I get nothing about that. I did understand what is an increasing function ...
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468 views

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 matrices and Algebra

Let us consider, a set of binary rectangular matrices of finite dimensions, call the set as $T$. The cardinality of the set $T$ is $2^{mn}$ where each matrix are of order m cross n. Suppose $S$ is a ...
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416 views

convert circuit to nor only gates

for an assignment I need to convert a circuit to NOR gates only circuit. (A+B)C + D I know that morgan's theorem states: (a) (A+B)'=A'B' (b) (AB)'=A'+B' I've seen online how to convert some ...
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Boolean Logic - Why doesn't the Associativity Law work in this scenario?

By the associativity law, shouldn't the statement below be true? I understand that the truth tables are different but where exactly does the associativity law apply then if this is False? ...
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Boolean Simplification (ABCD)' + ((CD)'(B+D)'

I have to simplify (ABCD)' + ((CD)'(B+D)' function using boolean algebra. I simplified it using a truth table and got ...
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Using rules of inference (Leibniz) to prove theorems.

Leibniz: If $A \equiv B$ is a theorem, then so is $C[p:= A] \equiv C[p:= B]$. Note: p is "fresh" means p doesn't occur in $A, B, C$. I am trying to understand how to use Leibniz rule of inference for ...
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Boolean equation simplification

This is the problem: XY’ + XYZ + XY'Z= X + Y'Z And so far I have this, XY’ + XYZ + XY'Z= X + Y'Z X(Y’ + YZ + Y’Z) Factor out X X(Y’ + Z + Y’Z) De Morgan Any tips on how to proceed? I know ...
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need help simplifying boolean algebra exrpressions

Can someone walk me through simplifying the following expression? $$a\lnot b\lnot s + ab \lnot s + \lnot abs + abs$$ help and advice is appreciated!
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Quotient of boolean algebra by an ideal

The homomorphism theorem states that every boolean ideal $I$ of a boolean algebra $A$ is the kernel of a boolean isomorphism. I'm reading a paper where the author presents a short proof of this ...
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Simplify this Logic Function?

Have a Hardware Lab to do, and I need to reduce the following function before I actually hook it up to the Logic Trainer. (not ac) + (abc) + (a not c) Or: $\lnot (a \land c) \lor (a \land b ...
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171 views

Simplification of a Boolean Expression

I want to simplify this expression: ACD' + E(A+C)(A'+D') + A'C . The result must be a product of sums, where every sum should be consisted of just two variables. For example (A+B)(C+A)(Z+Y) ... ...
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simplify the boolean expression

I'm fairly new too boolean algebra. I've tried simplifying this equation but I'm not quite sure if I've done it correctly. Simplify to 1 literal, (X + Y + Y'Z)(Y + X)(Y + X') My attempt: ...
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Prove $(A \backslash B)'\backslash (B \backslash A) = B \backslash A$

Let $A$ and $B$ be subsets of some universal set. Prove that $(A \backslash B)'\backslash (B \backslash A) = B \backslash A$ Given: Definition 3.3.1 states that $A$ and $B$ are sets. The complement ...