What is Boolean algebra: Definition and 153 Discussions
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively. Instead of elementary algebra, where the values of the variables are numbers and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction (and) denoted as ∧, the disjunction (or) denoted as ∨, and the negation (not) denoted as ¬. It is thus a formalism for describing logical operations, in the same way that elementary algebra describes numerical operations.
Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854).
According to Huntington, the term "Boolean algebra" was first suggested by Sheffer in 1913, although Charles Sanders Peirce gave the title "A Boolean Algebra with One Constant" to the first chapter of his "The Simplest Mathematics" in 1880.
Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages. It is also used in set theory and statistics.
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I want to prove by using the rules of boolean algebra that the following statement is always true $$\{b\land [\neg a\Rightarrow \neg b]\} \Rightarrow a$$
Since we have to use the rules of boolean algebra, we cannot use the truth table, right?
Could you give me a hint how we could...
Hello, can anyone help how i prove this logic rule? I am not sure whether i have to draw a digital circuit or something. if someone could help me solve it showing the steps they took i'd appreciate thanks
I was asked to use De Morgans law to find the complement of a particular equation, I applied the law correctly and simplified my solution down to
A'B'CD'+CA+CB'+D'A+D'B'
I ran the problem through a boolean simplifier to check my work...
Can anyone tell me if I'm doing this right so far? My question really is... does this mean that the next step for me is to expand the last statement shown?Asking because that seems like a lot of work.
I have received (unasked) a digital edition of "Laws of Form" (1969) by G. Spencer-Brown; I have glanced at it, and also at the Wikipedia article https://en.wikipedia.org/wiki/Laws_of_Form. OK, another logical system; logical journals (e.g. by ASL) are full of them, and I am not sure whether...
Hello to everyone who's reading this.
The problem I need help with is the following.:
Homework Statement
"Simplify to obtain minimum SOP.
F(A, B, C, D) = A’B’CD’+AC’D’+ABC’+AB’C+AB’C+BC’D"
The problem stated above has two provided solutions, the "main" one and the "alternate" one.
I'm...
Homework Statement
i'm viewing an example written in class. it looks like this:
f(x1, x2, x3, x4) = [(not x1) * x2 * x4] ∨ [x2 * x3 * x4]
what should be function after applying absorption law?
Homework Equations
i know how another option called "gluing" works:
[x1 * x2 * x3] ∨ [(not x1) *...
Homework Statement
My solution, is this correct?
This is what I came up with.
Y=A+((A*B)+B+C'+(B+C'*D)+D)
Is it safe to say that it is correct or did I make a mistake?
Homework Statement
The problem is given in the picture attached. It is a network of switches.Homework EquationsThe Attempt at a Solution
I managed to simplify the expression to this:
## (S + x'(w+y) + xvz)(x'+y)(v+z') ##
but I just can't find a way to simplify it to 9 literals. I've tried all...
Homework Statement
given the function
$F(a,b,c,d)=Σ(0,1,2,4,5,10,12)$
how can i know for sure if 8 is a don't care?
and is it possible for 6 and 14 to be don't cares?
Homework Equations
it's part of a bigger exercise, so the boolean expression i got from previous parts is a'c'+c'd'+b'd'
The...
Homework Statement
[/B]
How to simplife it? I would like to see a process of simplification.
Y = A'B'C'D + A'B'CD + A'BC'D + A'BCD' + A'BCD + AB'C'D + AB'CD + ABC'D'
The ' denotes a bar over the previous letter.My second question is:
Y = BC + AB'D = (uploaded picture)
Y = A'D + B'D + A'BC +...
Homework Statement
This excresice is supposed to help you understand the basic operations of sets, later used in probability. I am given the following phrases and have to write them in using mathematics.
Given three events A, B and C, which belong to sample space S, calculate the following...
For the expression:
(B and C) or (not B and not C) or (A and (not B) and not C)
I get two different answers depending on the order I do the simplification.
1. If I factor C out of the first and third terms I get (B and C) or (A and C) or (not B and not C)
2. If I factor not B out of the...
Homework Statement
This is from a past exam paper for logic design in my course. I have an exam coming up and would love to know how to solve this one.
[/B]
Develop a circuit realization of the XOR function with three inputs. You may use AND, OR and NOT-gates with not more than two inputs...
Homework Statement
Prove that $$(\bar{a} + b)(b+c) + a\bar{b}$$ where ##a,b## can be from the set ##B\in\{0, 1\}## equals $$a+b+c$$
Homework Equations
Rules of Boolean Algebra
3. The Attempt at a Solution [/B]
My attempt:
##\bar{a}b + \bar{a}c + bb + bc + a\bar{b}##
##b(\bar{a} + 1+c) +...
Homework Statement
Express the function Y= (abd + c)' + ((acd)'+(b)')' as the complete disjunctive normal form:
2.1 by applying Boole's theorerm,
Homework EquationsThe Attempt at a Solution
I separated the equations to two terms (T1,T2)
T1= (abd + c)' T2=((acd)'+(b)')'
T1= (abd+c)'...
The problem
I have been trying to solve a long problem but my answer differs from my books answer with just a few peculiar terms.
My answer:
##x_1' \vee x_0'x_1x_2 \vee x_0x_1'x_2' \vee x_2##
Book:
##x_1' \vee x_2 ##
My question is:
Is it possible to simplify ##x_0'x_1x_2 \vee...
Homework Statement
Use the definition of exclusive or (XOR), the facts that XOR commutes and
associates (if you need this) and all the non-XOR axioms and theorems you know
from Boolean algebra to prove this distributive rule:
A*(B (XOR) C) = (A*B) (XOR) (A*C)
Homework Equations
All the...
The problem
This is not a complete homework problem. I am at the last step of the solution to a long problem and only interested to know whether these following expressions are equivalent.
My answer:
## a \oplus ab \oplus ac ##
Answer in my book:
## a \oplus b \oplus c ##
The attempt
I...
The problem
I am trying to show that ##a'c' \vee c'd \vee ab'd ## is equivalent to ## (a \vee c')(b' \vee c')(a' \vee d) ##
The attempt
## (a \vee c')(b' \vee c')(a' \vee d) \\ (c' \vee (ab'))(a' \vee d)##
The following step is the step I am unsure about. I am distributing the left...
I am asking this question as I have found Boolean Algebra quite intriguing. I have a good understanding of high school level probability and statistics and also Algebra II. Is this enough or do I need more "mathematical maturity"? Anyway, thank you in advance.
Homework Statement
(A OR C) AND NOT(C AND A AND B OR C AND A AND NOT B)
or
(A + C) (CAB + CAB')'
Relevant Equations
(A+B)' = A'B'
A(B+C) = (AB) + (AC)
(AB)' = A' + B'
The attempt at a solution
I'm not sure how I'm suppose to expand (CAB + CAB')' for simplifying. I keep arriving at false which...
https://scontent-mxp1-1.xx.fbcdn.net/v/t35.0-12/13324008_10209600482268675_3779883_o.png?oh=abd4c815b9472f70cfbfbb1923a37b3c&oe=5752F454
Number 12 : I have simplified it and I have got AB but the model answer in red say A(comp(B)+c)
I just want is that answer true or not , thanks in advance
Homework Statement
I'm trying to show that the output of this XOR circuit is ##F=A'B+AB'##,
Homework Equations
##(A+B)'=A'\cdot B'##
##(A\cdot B)'=A'+B'##
The Attempt at a Solution
From the gates the output is ##[(A\cdot B)+(A+B)']'##, using De Morgan's laws this becomes ##[(A\cdot...
This truth table that represents statement p v (q ^ r) is equivalent to (p v q) ^ (p v r)Showing that this statement is not equivalent to (p v q) ^ r.. Now I need to what property of Boolean Algebra is being demonstrated by the fact that the first two statements were equivalentp q r q ^ r...
Most of the results on google happily prove A+(B.C) = (A+B).(A+C), which is that OR is distributive (over AND).
But as part of their proof, they use the law that AND is distributive (over OR), namely that
A.(B+C) = (A.B)+(A.C) which I can't seem to find any algebraic proof for.
So are there...
Homework Statement
Am I doing this right? The question is...
(ab) + (a' + b') = 1
Homework Equations
(a) Commutative a · b = b · a a + b = b + a
(b) Associative (a · b) · c = a · (b · c)...
So, I have an equation:
~A * B * C * ~D + ~A * B * C * D + A * ~B * ~C * D + A * ~B * C * D + A * B * ~C * D + A * B * C * ~D
where * represents "AND" and + represents "OR", ~ being NOT.
Part of the reason I'm having trouble is due to the length of the equation.
So far, I've managed to use...
Homework Statement
Minimize the following using boolean identities
1. AB'CD+(ABC')'+ABCD'
Homework Equations
Identity 1A=A 0+A = A
Null (or Dominance) Law 0A = 0 1+A = 1
Idempotence Law AA = A...
I know De-Morgan's law that $$ -(p∧q) = -p∨-q $$
Also $$ -(p∨q) = -p∧-q $$
But for material implication and bi conditional operations there are also some transformation.
What is the law or proof for it? Like
$$ p⇒q = -p∨q $$
$$ p ↔q = (p∧q) ∨ (-p∧-q) $$
There may be other properties also that I...
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >
(A+B)&(C+D) + (A+B)&(C+D)' + C
(A+B)&(C+D) + (A+B)&(C'&D') + C by deMorgans
(A+B)&[(C+D)+(C'&D')] + C by Distributive
I'm just wondering if I did anything wrong in this simplification or if it...
Homework Statement
In Game 6 of a 6-team double-elimination tournament, Team 1, the top-ranked team, faces the loser of a previous game involving Team 2, the second-ranked team, against one of either Team 3, the third-ranked team or Team 6, the sixth-ranked team.
This exercise tests the limits...
C) How did they go from the first red line to the second?
f) How did they go from the first green line to the second
g) B + B(bar) = 1, so surely the answer should have a +1 ?
2) How did they go from the first purple line to the second?
I have a list of the Boolean laws and I have used...
Homework Statement
Simplify (A+B')(B+C)
The Attempt at a Solution
I first expanded it and got
= AB + AC + B'B + B'C
= AB + AC + B'C
Turns out the solution is AB + B'C (according to an online source). How do we get rid of the AC term?
Homework Statement
I need to obtain the minimum second order circuit of the function:
Homework Equations
The Attempt at a Solution
I know that in order to get the minimum second order circuit, I need to use K-Map using both miniterms and maxiterm and compare them. My problem is to put the...
Homework Statement
Homework Equations
Boolean Algebra
The Attempt at a Solution
I use the distri to change the A+C'.D and Demorgan.
Should i use dis and Demorgan firstly like below?
Which property I should use in the next step?
If the first part is wrong,which property i should...
According to wikipedia, absorption is an axiom for a boolean algebra. This seems incorrect to me, since I believe absorption can be proved from the other axioms (distributivity, associativity, commutativity, complement, identity).
Thoughts?
## AB' + A = AB' + A*1 = A(B'+1) = A(1) = A ##
BiP
Homework Statement
Prove the following expression using Boolean algebra:
1. X'Y' + Y'Z + XZ + XY + Z'Y = X'Y' + XZ + YZ'
Homework Equations
Laws of Boolean algebra
The Attempt at a Solution
I tried to take Y common but failed. I did the same with X and Z, but the method did...
I am a bit confused about a question on proving partial order relation. here is the question and what i done so far.
"define the relation '≤' on a boolean algebra B by
for all x,yεB x≤y if and only if xVy=y, show that '≤' is a partial order relation"
first of all what exactly does...
Homework Statement
I'm studying function simplification in boolean algebra, and I didnt understand the following step:
(NOT A)(NOT B)(C) + B = (NOT A)(C) + B
What happened to the NOT B?
Homework Equations
The Attempt at a Solution
Hi I am not sure where to post this question but I am trying to simplify this expression:
r*c'w+c (As in R AND NOT C AND W OR C) to c+wr (As in C OR W AND R) and I know that it simplifies to this and they are both equivalent; however my question is which boolean simplification property is...