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

learn more… | top users | synonyms

0
votes
2answers
41 views

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 ...
3
votes
2answers
287 views

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 ...
10
votes
2answers
321 views

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: ...
2
votes
0answers
38 views

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 ...
3
votes
1answer
85 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? $$abc * a'bc + (abc)' * (a'bc)'\quad = \quad \;b'+c'$$
2
votes
1answer
61 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 ...
1
vote
1answer
3k views

How does it evaluate A XOR B XOR C?

I am trying to solve the following combination, ...
0
votes
1answer
43 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 ...
1
vote
1answer
34 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.
2
votes
3answers
85 views

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 ...
2
votes
1answer
97 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 ...
1
vote
1answer
53 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!
0
votes
0answers
41 views

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 $ ...
0
votes
1answer
26 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 ...
0
votes
1answer
93 views

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 ? ...
0
votes
2answers
592 views

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 ...
2
votes
0answers
120 views

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, ...
0
votes
2answers
657 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?
10
votes
1answer
205 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 ...
1
vote
1answer
132 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 ...
2
votes
2answers
128 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 ...
2
votes
1answer
150 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 ...
3
votes
1answer
71 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: ...
2
votes
1answer
61 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 ...
1
vote
1answer
57 views

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.
0
votes
1answer
232 views

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.
1
vote
1answer
2k views

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 ...
1
vote
1answer
87 views

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’) = ...
2
votes
2answers
32 views

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!
1
vote
1answer
508 views

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 ...
0
votes
1answer
140 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?
0
votes
1answer
112 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? ...
0
votes
1answer
86 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
0
votes
1answer
167 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)?
2
votes
2answers
42 views

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 ...
2
votes
2answers
110 views

Proving if Boolean Equations are valid

I need to prove algebraically that: $$ab + abc'd + abde' + abc'e + a'b = b$$ $$(wxyz)(wxyz' + wx'yz + w'xyz + wxy'z) = 0$$
1
vote
0answers
114 views

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 ...
1
vote
1answer
87 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 ...
0
votes
1answer
253 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 ...
4
votes
3answers
7k views

De-Morgan's theorem for 3 variables?

The most relative that I found on Google for de morgan's 3 variable was: (ABC)' = A' + B' + C'. I didn't find the answer for my question, therefore I'll ask here: ...
0
votes
2answers
241 views

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 ...
1
vote
2answers
68 views

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?
1
vote
1answer
880 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 ...
2
votes
2answers
65 views

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 ...
3
votes
1answer
724 views

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 ...
-1
votes
1answer
169 views

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 ...
2
votes
5answers
140 views

Value of $(a=1) \wedge (b=1) \wedge (c=2)$ given $a=1$, $b=2$ and $c=2$

How would I solve the following question. Assume $a=1$, $b=2$ and $c=2$ what is the value of the following Boolean expression $(a=1)$ AND $(b=1)$ AND $(c=2)$ I am kind of confused because I know ...
0
votes
3answers
380 views

Simplifying Boolean Algebra

I am trying to prove that BC + !A!B + !A!C = ABC +!A I have attempted using De Morgan laws, and substituting X for !A!B and ...
-1
votes
1answer
211 views

Boolean Algrebra: Karnaugh Map [closed]

Using the Karnaugh map, express the following function: F(0, 1, 4, 5, 8, 10, 11, 12, 13, 15)
0
votes
1answer
76 views

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