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 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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Karnaugh map question $\displaystyle \sum_{} (2,3,6,7) $

I can't write here the map of Karnaugh of this function so I just ask whether this reduction goes after Y? $F(w,x,y) =\displaystyle \sum_{} (2,3,6,7) $ In addition if there are two functions like: ...
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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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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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Write the following functions in algebraic sum of multiples or multiplying Amounts. If possible,simplify the expression

Write the following functions in algebraic sum of multiples or multiplying Amounts. If possible,simplify the expression The Question is: $F(A,B,C): Maxterms(4,5,6,7)$ : $M4 = 100 => A'+B+C $ ...
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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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Memory and bits. Need some help

Could someone check over my answers to verify I am correct. Say we have a memory consisting of 2048 locations, and each location contains 16 bits. ◦ A) How many bits are required for the address? ...
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Free algebra (Boolean algebra)

Could someone give me a simple explanation of Free Algebra on $\kappa$. How to construct free($\omega$). here is it says http://en.wikipedia.org/wiki/Free_Boolean_algebra free($\omega$) is equal to ...
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How to prove that $(A \lor B) \land (\lnot A \lor B) = B$

I know this is fairly basic, and I understand that it becomes $$ \begin{align} (A \land \lnot A) \lor B \\ F \lor B \\ B \end{align} $$ However, I can't work out how to prove that it becomes that ...
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How do I find the size of this set?

For homework, I need to show that the size of a certain set is $\le 2^{(3n)^k}$ but I'm not getting this (I think I may just misunderstand how the set is defined). So the set is defined as follows: ...
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Boolean Algebra Question

my problem is ,Please give the algorithm: how can rewrite an arbitrary propositional formula alfa(α) into a proposional formula beta(β) so that beta does not contain disjunction(∧) and alfa ...
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Is it true “Every Boolean algebra is an algebra of sets, for any given set X”

I have confused between these two notions, please help Every Boolean algebra is an algebra of sets, for any given set $X$, and the converse is false.
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Proof of Associativity in Boolean Algebra

I must prove the most basic associativity in boolean algebra and there is two equation to be proved: (1) a+(b+c) = (a+b)+c (where + indicates OR). (2) a.(b.c) = (a.b).c (where . indicates AND). I ...
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Trouble understanding boolean logic proof.

*Find the complement of $F=x+yz$; then show that $FF’ = 0$ and $F + F’ = 1$ $F(x,y) = x+yz$ $F’(x,y) = (x+yz)’ = x’(yz)’ = x’(y’+z’)$ $FF’ = (x+yz)x’(y’+z’) = (xx’+x’yz)(y’+z’) = x’yz(y’+z’) = ...
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Get A⊕(B+1) from A⊕B

I have numbers A,B,C.D. (⊕ is XOR) C = A⊕B D = A⊕(B+1) Is there any way to get D from C, when I do not know A and B? How? Thanks for help!
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Implement using only XOR gates F=A'B'C'D+A'B'CD'+A'BC'D'+A'BCD+AB'CD

How can we implement the function: F=A'B'C'D+A'B'CD'+A'BC'D'+A'BCD+AB'CD without simplifying it and using ONLY XOR gates (not using AND/OR gates) ? NOT gates are usable too, since they can be ...
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125 views

Software to Find Kernels/Co-Kernels of Boolean Expressions

Is there any (free) software available that calculates all the possible kernel/co-kernel pairs of a boolean expression?
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LC-3 instruction set. Help needed

Using only one LC-3 instruction, how would I move the value in Register 2 into Register 3 How to perform R1 = R2 - R3 using only 3 LC-3 instructions? Hope you can help. Thanks
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Memory and bits

If a memory's addressability is 64 bits. What does that tell you about the size of the memory address register (MAR) and memory data register (MDR)?
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finding signs of 3 numbers

This is slightly more of a coding problem than a math problem but I think it is still relevant. So let's say I have 3 numbers A,B,C and I can only call a given function if two are negative and one ...
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Proving if Boolean Equations are valid

I need to prove algebraically that: $$ab + abc'd + abde' + abc'e + a'b = b$$ $$(wxyz)(wxyz' + wx'yz + w'xyz + wxy'z) = 0$$
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Showing that a Boolean algebra is a Boolean ring

I've proved that a Boolean ring is a Boolean algebra but I am having trouble with the converse. The operation for + is defined as the symmetric difference for elements $a$ and $b$ from the Boolean ...
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84 views

Simplify $F=MNO+Q'P'N'+PRM+Q'OMP'+MR$

How can we simplify $$F=MNO+Q'P'N'+PRM+Q'OMP'+MR$$ using the theorems of boolean algebra, not Karnaugh or anything else? Well, I can obviously simplify $MR(P+1)=MR$, so the expression becomes ...
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199 views

Is there a unique minimal expression for every boolean function?

Is there a unique minimal expression for every boolean function? I've heard that there are some boolean expressions for which the minimal form is not unique. What are the characteristics of this kind ...
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153 views

Infitive distributive law in boolean valued models

I'm posting the problem 2.14 and 2.15 of the book "Set theory" of J.L. Bell. These problem are proposed after the forcing relation chapter and I'm new in this kind of stuff, so I have some little ...
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Binary Operations on Subsets--Two Questions

I have two questions about the properties of binary set operations that I am having difficulty arriving at answers that I completely trust (though I am sure they are not difficult questions). Here ...
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Partially Ordered Sets question

For $m\in\mathbb{N}$, which integers are covered by $m$? I've been playing with the prime factors of $m$ and I can't seem to see any pattern. Can anyone help?
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How can I prove this logical expression?

I've already confirmed that the following expression is true with a truth table, but I need to prove this with other Boolean expressions for my assignment. The $\oplus$ symbol is exclusive or in this ...
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803 views

Reduce following expression to one literal, boolean algebra

$$W'X(Z'+Y'Z)+X(W+W'YZ)$$ The goal is to reduce the following to one literal So after I expanded it out, i got the following: $$W'XZ'+W'XY'Z+WX+W'XYZ$$ Now from here, I got stuck and didn't know ...
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Designing a circuit: Hamming Code

How would I design and build a circuit that would generate check bits for 4-bit word? In this instance, the same circuit should also be used to generate check bits for when you read data back in case ...
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Desiging a circuit that implements Hamming Code

How would I design and build a circuit that would generate check bits for 4-bit word? In this instance, the same circuit should also be used to generate check bits for when you read data back in ...
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201 views

Boolean Algrebra: Karnaugh Map

Using the Karnaugh map, express the following function: F(0, 1, 4, 5, 8, 10, 11, 12, 13, 15)
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Boolean Algebra (Help Needed)

How would I draw the gate-level logic circuit of the following Boolean expression? $$ (((A \land B \lor C) \lor D \land E \land F) \lor G \land (H \lor I \land J)). $$ Then how would I implement this ...
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How to show Boolean identity : $(ab + c + d)(c' + d)(c' + d + e) = abc' + d$

How to show Boolean identity : $(ab + c + d)(c' + d)(c' + d + e) = abc' + d$
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Boolean Algebra proving algebraically simple

$$(X'+Y )(X+Y')=XY+X'Y'$$ I am just wondering how these are equal, and what laws are used to get there
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Conjunctive normal form of logical expression

I tried to convert this to a CNF-expression but failed. What did I do wrong? Or are there simply missing steps? $$ F' = (( A \lor \lnot B) \land C) \to ( \lnot A \land C) $$ Removed Implication $$ ...
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Prove the identity in this boolean equation

$$AD'+A'B+C'D+B'C=(A'+B'+C'+D')(A+B+C+D)$$ Don't know where to begin with this.
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How do we know what $A$ or $B$ or $C$ is after simplifying?

I understand the basics of boolean algebra and how to simplify them. What I am confused about is how do we know what to call a value after simplifying? Imagine we create a boolean algebra expression ...
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Boolean Algebra, 4-variable Expression Simplification

I have the following Boolean expression: $$w'x'y'z + wx'y'z + xz + xyz'\tag{1}$$ Upon doing my own work, I can only get as far as: $$zx + xy + zy'\tag{2}$$ Now, when I put the original equation ...
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Logic gates analyses

How to write the output of the gates not, and, or, xor, nand and nor in terms of their inputs, expressed as zeros and ones, using base 10 addition and multiplication. Thanks much in advance!!!
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Boolean Algebra - Product of Sums

I converted from a truth table to sum of products and simplified that easily. What I am having problems with is simplifying the product of sums for that same truth table. I have: NOTE: $A' = ...
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Incompleteness of Connectives

I’m currently trying to learn more about Mathematical Logic and have reached a sticking point. I also have the solutions to the problems I’m working through and I usually don’t need to ask for help to ...
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203 views

Can anyone Simplify this Boolean expression?

The expression is: [AB {C+(BD)'} + (AB)']CD
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How can I express xNORy solely with NAND operations?

I've tried every which way I can think of to manipulate the algebra using the various laws I was given, but I cannot figure out a way to get $\overline{x+y}$ to convert to only NAND operations using ...
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Atomic Boolean lattice is weakly atomic

The book "Introduction to Lattices and Order" says at Exercise 10.12 that an atomic Boolean lattice is weakly atomic. Could you tell me why it holds?
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Construct countable Boolean algebra

How can I construct a countably infinite Boolean algebra with $n$ atoms, for $n \in \mathbb{N}$?