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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  1. M

    A What are your insights on The Hardest Logic Puzzle Ever?

    Hello! I’m an assistant of a mathematical scientific researcher, and my research programme evolves around finding and developing all the (possible) solutions regarding all unsolved mathematical, logic, exact, and IQ puzzles ever created. If you search on the internet for: “The hardest unsolved...
  2. M

    MHB Prove that the statement is always true using the rules of boolean algebra

    Hey! 😊 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...
  3. M

    Comp Sci Proving a Logic Rule with Boolean Algebra: Step-by-Step Guide

    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
  4. garthenar

    Comp Sci Boolean simplification of a larger term?

    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...
  5. R

    Engineering Prove using Boolean algebra that two expressions are equivalent

    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.
  6. M

    MHB Two-element Boolean algebra: How are the equalities derived?

    Hey! :o We have the following equalities: \begin{align*}\left (\overline{y}\land z\lor x\land \left (\overline{z}\lor y\right )\right )\land \left (\overline{x}\lor \overline{y}\right )& \overset{(1)}{=}\left (\overline{y}\land z\lor x\land \overline{z}\lor x\land y\right )\land \left...
  7. nomadreid

    I "Laws of Form" by G. Spencer-Brown (1969)

    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...
  8. S

    Boolean Algebra, Minimum Sum of Products Problem

    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...
  9. L

    Boolean algebra absorption law

    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) *...
  10. M

    (Boolean Algebra) Did I write this logic expression correctly?

    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?
  11. toforfiltum

    Engineering Simplifying switching circuit literals

    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...
  12. E

    How to know if a certain number is a don't care in KMAP

    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...
  13. R

    Boolean Algebra Simplification

    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 +...
  14. Sollicitans

    Calculating events from phrasal expressions

    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...
  15. Charles Henderson

    Boolean Algebra - Simplifying to two different expressions

    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...
  16. N

    Engineering Circuit Realization of XOR function with Three inputs

    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...
  17. doktorwho

    Can you help me solve this Boolean algebra problem?

    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) +...
  18. A

    Need to be sure of this boolean algebra problem's solution

    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)'...
  19. Rectifier

    Boolean algebra - is it possible to simplify this expression to 0

    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...
  20. PhotonSSBM

    Boolean Algebra Proof (Distribution and XOR)

    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...
  21. Rectifier

    The Boolean Algebra XOR Problem: Are These Expressions Equivalent?

    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...
  22. Rectifier

    Boolean algebra - distribution

    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...
  23. Mikaelochi

    What are the prerequisites for learning Boolean Algebra?

    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.
  24. J

    Simplifying Boolean Algebra: How to Simplify Complex Boolean Expressions

    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...
  25. S

    MHB Proving A∪B=A∩B iff A=B with Boolean Algebra

    I want to prove: A\cup B=A\cap B\Longleftrightarrow A=B Forall A,B sets By using the axioms and theorems of the Boolean Algebra. Any hints ??
  26. waver.

    Boolean Algebra Problem: Solution for AB vs. A(comp(B)+c)

    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
  27. Kernul

    Solve Boolean Algebra Homework

    Homework Statement Simplify the expression ##cb' + ca'b + cabd + cad'## Homework Equations All the properties of boolean algebra. The Attempt at a Solution Here's how I did it: $$cb' + ca'b + cabd + cad' = $$ $$c(b' + a'b + abd + ad') = $$ $$c(b' + a'b + a(bd + d')) = $$ $$c((a + a')b' + a'b...
  28. P

    Simplifying output for a XOR gate using Boolean Algebra

    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...
  29. barbara

    MHB Demonstrating Distributive Property of Boolean Algebra

    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...
  30. Valour549

    Prove A.(B+C) = (A.B)+(A.C) <Boolean Algebra>

    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...
  31. M

    Understanding Boolean Algebra: Homework Help and Examples

    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)...
  32. T

    Simplifying a 4-term equation using boolean algebra

    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...
  33. Duderonimous

    How Can Boolean Identities Simplify AB'CD+(ABC')'+ABCD'?

    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...
  34. P

    Can Boolean Algebra Simplify Complex Expressions Using Postulates and Theorems?

    1. Simplify the expression: F = xyz' + xy'z' + x'yz + xyz2. Postulates and theoremsThe Attempt at a Solution F = xyz' + xy'z' + x'yz + xyz = x(yz' + y'z') + yz(x' + x) (Distributive) = x(yz' + y'z') + yz.1 (Complement) = x(yz' + y'z') + yz (identity) This is where I need...
  35. P

    How to simplify the expression -- Boolean algebra

    1. prove that: X'Y'Z + X'YZ' + XY'Z' + XYZ = (X⊕Y)⊕Z Homework Equations Use postulates and theorems. The Attempt at a Solution X'Y'Z + X'YZ' + XY'Z' + XYZ (original expression) X'Y'Z + X'YZ' + X(Y'Z' + YZ) (distributive) X'Y'Z + X'YZ' + X.1 (complement) X'Y'Z + X'YZ' + X (identity)[/B]...
  36. R

    Transformation rules in Boolean algebra

    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...
  37. K

    Boolean Algebra Simplification

    ab'c + a'b + bc' + abc = ac + a'b + bc' (How to further reduce this?) Kmap gives B + AC
  38. Extreme112

    Help with simplifying boolean expression

    < 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...
  39. J

    Calculating Probabilities in a Double-Elimination Tournament

    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...
  40. I

    Boolean Algebra Identities: How do they work?

    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...
  41. T

    Digital logic - Boolean algebra simplification problem

    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?
  42. T

    Boolean Algebra Question (Digital Logic Circuits)

    Homework Statement The Attempt at a Solution Is my solution correct or can I simplify it even further?
  43. F

    Help putting expression in form of miniterms (Boolean algebra)

    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...
  44. C

    Simplifying Boolean Algebra Equations with Distribution and DeMorgan's Law

    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...
  45. B

    Is absorption an axiom for a boolean algebra?

    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
  46. H

    Boolean algebra prrof question

    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...
  47. B

    Partial order relations, on boolean algebra

    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...
  48. R

    Why (NOT A)(NOT B)(C) + B = (NOT A)(C) + B [Boolean Algebra]

    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
  49. B

    Boolean Algebra Simplification Property Question

    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...