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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Is it possible to derive all the other boolean functions by taking other primitives different of $NAND$?

I was reading the TECS book (The Elements of Computing Systems), in the book, we start to build the other logical gates with a single primitive logical gate, the $NAND$ gate. With it, we could easily ...
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Logic Circuit Question

1) Write the boolean expression after every GATE 2) Write the boolean expression of GATE 3 3) Try to simplify the boolean expression of GATE3 I need to know if what I did its right + your advice ...
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Boolean simplification, 5 variables

I'm currently learning for my maths exam, and in the part about boolean algebra I came across an exercise that I can't seem to solve. I probably only need the first few steps to get started. $$ (xyz ...
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Simplify Boolean Algebra

How do I simplify the following expression with Boolean Algebra? Please show what you used to simplify so I can understand. $$ABC + AB'C' + ABC' + A'B'C'$$
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Duality principle in boolean algebra

All the definitions I came across so far stated, that if a statement is true, then also its dual statement is true and this dual statement is obtained by changing + ...
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maximal antichain

I don't understand the definition of Jech (set theory) for "maximal antichain". Let $B$ a boolean algebra and $A$ a subalgebra of $B$. $W\subseteq A^+$ is a maximal antichain if $\sum W=1$ and $W$ ...
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Why is $ab + bc + c\bar{a} = ab + c\bar{a}$ true in binary?

I was simplifying the equation of a logic gates problem and I realized that ab + bc + cā and ab + cā followed the same truth ...
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45 views

completeness and saturation

Let $B$ a complete Boolean algebra. Suppose, for $\kappa$ cardinal, that $B$ is not $\kappa$-saturated. Then there exists a partition $W$ of $B$. Because of completeness, we have $B=\sum W\in B$. So ...
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Which law of logical equivalence says $P\Leftrightarrow Q ≡ (P\lor Q) \Rightarrow(P\land Q)$

I'm going through the exercises in the book Discrete Mathematics with Applications. I'm asked to show that two circuits are equivalent by converting them to boolean expressions and using the laws in ...
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108 views

Isomorphism Subalgebra

Given, the unit interval $I$ endowed with the Lebesgue measure $\mu$, and let $A$ be the (Boolean) algebra of Jordan measurable subsets of $X$ with respect to $\mu$, (i.e. those sets that satisfying ...
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533 views

Rationale behind truth values

I originaly asked a question on Programmers.SE to know why $0$ was consider $\text{false}$ and all the other [integral] values were considered $\text{true}$. That was a huge debate and many said it ...
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Generalization of Boolean OR?

I have been looking at the Boolean OR function and Im trying to find its integral analogue. What I mean is: Boolean AND (x, y) where x and y are Boolean Values with 0 = False, 1 = True is equivalent ...
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Reducing Boolean expressions

Just learning mathematical proof writing and came upon this interesting question Writing an expression using logic. $$(P \land Q \land \lnot R) \lor (P \land \lnot Q \land \lnot R) \lor (\lnot P ...
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A matrix w/integer eigenvalues and trigonometric identity

Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated: Let $n$ be a natural number. (a) Consider the following Toeplitz/circulant symmetric matrix: ...
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Inferring simplest method to convert bit array 1 to bit array 2.

Consider the set of all bit arrays of length $n$. Now consider the set of all 1-to-1 functions that map from this set to this set. Now select a single function out of the latter set. Is there any ...
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1answer
133 views

Boolean Algebra Transform

I am revisiting Boolean algebra after a long while. Can somebody help show me how to simplify the LHS to get the RHS? $$abc * a'bc + (abc)' * (a'bc)'\quad = \quad \;b'+c'$$
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1answer
64 views

Filters of Boolean algebras which are Boolean algebras

Looking at some filters generated by elements of a finite Boolean algebra I have the impression that many/most/all of them are by themselves Boolean algebras (at least I didn't stumble upon a ...
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1answer
5k views

How does it evaluate A XOR B XOR C?

I am trying to solve the following combination, ...
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46 views

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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37 views

Boolean Equation Transformation

Can someone show me the steps in getting from $f = (ab + c')(d' + e + f')$ to $f = abe +ab(df)' + c'e + c'(df)'$? I am trying to relearn Boolean algebra after a long hiatus.
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Prove that $(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$.

Logically, the following proposition makes sense: $(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$ Or, in english, if sets $S$ and $T$ share no elements, and the union of ...
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1answer
54 views

truth table for followings

Hi I am new to this site. I got an assessment to complete tomorrow. Its about Computer programming, and i am having trouble with these questions. Can anyone please help me. Using truth tables show ...
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1answer
112 views

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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1answer
59 views

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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1answer
39 views

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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898 views

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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1answer
216 views

Stone's Representation Theorem and The Compactness Theorem

If you're working on $\mathsf {ZF}$ and you assume the compactness theorem for propositional logic, then you have the prime ideal theorem, and thus you can show that the dual of the category of ...
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1answer
165 views

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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2answers
177 views

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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1answer
177 views

Can boolean homomorphisms of boolean algebras correspond to ultrafilters?

I am trying to solve 5th problem in Exercises 2.9 in Awodey's book on page 55: Show that for any boolean algebra $B$, boolean homomorphisms $h : B \to 2$ correspond exactly to ultrafilters in $B$. I ...
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1answer
79 views

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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1answer
64 views

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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1answer
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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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How can you design a 3 bit adder using a 4 bit adder?

How can you design a 3 bit adder using a 4 bit adder? The description and/or the circuit's scheme would be great.
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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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186 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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220 views

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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1answer
153 views

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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689 views

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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1answer
99 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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479 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 ...