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. I

    Boolean algebra -Sum of products form - theorem

    Homework Statement Show the following is true using theorems. State which theorem you use at each step. A'BD' + BCD + ABC' + AB'D = A'BD' + BCD + ABC' + AB'D + BC'D' + A'BC + ABD Homework Equations The Attempt at a Solution I am very new to boolean algebra and very...
  2. G

    Boolean algebra P-O-S => S-O-P

    Homework Statement Obtain a sum of products expression for the following: (A+B)(A+C')(A+D)(BC'D+E) Homework Equations .. The Attempt at a Solution Should I use DeMorgan's Law or can I simply expand the expression and then simplify? If I use DeMorgan's Law, won't I be...
  3. G

    A smal problem with boolean algebra

    There's something I can't seem to figure out. say we have the function f=xz +x'z' why is this equals (x xor z)' ? i thought using De-Morgan's law we shall get (x xor z)'=(x'z+z'x)'=(xz'zx')=0 then why does (x xor z)'=xz+x'z' ? and one more small thing, say we have (x xor y)...
  4. J

    Solving Boolean Algebra Qns: x+yz+x'y= (x+x')(x+y)+yz

    hi all ive got a short qns on boolean algebra. here's the qns x+yz+x'y= (x+x')(x+y)+yz this is just a small part of the eqn i do not understand. how does x+x'y become (x+x')(x+y)? i don't seem to be able to apply any other laws that lead to this. thanks in advance!
  5. E

    Solve Boolean Algebra with Help | 65 Chars

    1. Homework Statement Simplify the following expressions using Boolean Algebra. a) A + AB (A or A and B) b) A'BC + AC (A(not) and B and C or A and C) c) A'B + ABC' + ABC (A(not) and B or A and B and C(not) or A and B and C) 2. Homework Equations Some simple boolean algebra rules...
  6. E

    Simplifying Expressions with Boolean Algebra - Homework Help

    Homework Statement Simplify the following expressions using Boolean Algebra. a) A + AB (A or A and B) b) A'BC + AC (A(not) and B and C or A and C) c) A'B + ABC' + ABC (A(not) and B or A and B and C(not) or A and B and C) Homework Equations Some simple boolean algebra rules. x+1=0...
  7. O

    XOR gate to XNOR gate boolean algebra

    Xor gate with negated input and negated output. The expected output is an XNOR gate. I can't get it with boolean algebra. My initial formula is: ((AB')' + (A'B)')' The expected output is: AB + (AB)' right? Please help me.
  8. R

    Simplifying Boolean Expressions using the Laws of Boolean Algebra

    EDIT: My apologies if I started this thread in the wrong forum! Homework Statement Z = P.Q.R + P.R Where . = AND + = OR ~ = NOTHomework Equations (Commutative Law) A+B = B+A, A.B = B.A (Associate Law) (A+B)+C = A+(B+C), (A.B).C = A.(B.C) (Distributive Law) A(B+C) = AB+AC...
  9. M

    Simplify Boolean Algebra: A'C' + A'D' Solution Example

    Homework Statement [PLAIN]http://img340.imageshack.us/img340/7690/123rk.gif Homework Equations The answer should be A'C' + A'D'The Attempt at a Solution Shown in above in the question. I want to know how can i get the correct answer. As you see above, in my last step, I wonder whether B'+BD...
  10. 0

    Exploring the Mysteries of Boolean Algebra

    http://img710.imageshack.us/img710/2314/booleanalgebra.jpg AFAIK logic is all about "T"/"F" or 0/1, and boolean algebra is all about logical manipulation. But there seems to be something wrong since there is a boolean algebra with more that 2 objects in it`s set. So, can I have some...
  11. O

    Boolean algebra- cancellation property of addition

    1. Prove that for all boolean algebras if x+y=x+z and x'+y = x'+z then y=z. 2. Homework Equations : x+x' = 1, xx'=0, basically we are allowed to use the usual boolean algebra properties. 3.Attempt: This the second part of a problem, in the first part we had to give and example of...
  12. P

    Can you Simplify These Boolean Expressions Using the Laws of Boolean Algebra?

    1. Homework Statement and attempted solutions Simply each of the following Boolean expressions: i) http://img69.imageshack.us/img69/4438/equa1.jpg ii) http://img69.imageshack.us/img69/9907/equa2.jpg iii)Expand the following SOP form to proper miniterms (don't forget to delete...
  13. N

    Simple proof of Boolean Algebra

    Homework Statement Prove (x+y)(y+z)(x'+z) = (x+y)(x'+z) Homework Equations The Attempt at a Solution On LHS, i have done till =(y+xz)(x'+z) 1st- distributive =x'y+yz+xz 2nd- distributive(2ce) =(x+y)(x'+z) 3rd-? For third procedure, what properties should be...
  14. B

    Boolean algebra theorem question

    Homework Statement My book contains this boolean algebra theorem: A + notA * B = A + B I have verified the validity of this statement using truth tables, but I find that I am unable to derive it. Our professor gave us a few steps to simplify Boolean expressions: 1) change all...
  15. N

    Can Boolean Algebra Be Simplified to Just XY + Z?

    Homework Statement F(x,y,z)=X'Y'Z + X'YZ+ XY'Z + XYZ' + XYZ The Attempt at a Solution F(x,y,z) = X'Y'Z + X'YZ+ XY'Z + XY (Z'+Z) F(x,y,z) = X'Y'Z + X'YZ+ XY'Z + XY I am stuck here l don't know how to further simplify it so that l can end up with F= XY + Z
  16. O

    How do I simplify this boolean function using the rules of boolean algebra?

    Okay, so I'm stuck at a step of a much bigger problem where I have to simplify a boolean function. Homework Statement Here is where I'm stuck, I have to change the left hand side to the right hand side. How do I prove this with the rules of boolean algebra? CA + CB + B'A = CB + B'A...
  17. N

    Boolean algebra conceptual questions

    Guys I need your help! :cry: I've never liked boolean algebra, with conjugation, disjunction, implication, equivalence and all that stuff because I considered very easy to understand. But what I can not understand is implication (I get headaches from implications :frown:) Here is what I do...
  18. I

    Boolean Algebra: Minimum Sum-Of-Products Expression

    Homework Statement Find the minimum Sum-Of-Product Expression for: f = ab'c' + abd + ab'cd' The Attempt at a Solution By introducing the missing variable in term 1 and term 2 I can get an expression that has all the variables: a, b, c, and d. I do so by: f = ab'c'd + ab'c'd' + abcd + abc'd...
  19. I

    UTT- Boolean Algebra Equivalence

    Ok, I am having a difficult time with the following problem. I am to show that the two sides are equal. x1'x3 + x1x2x3' +x1'x2+x1x2' = x2'x3+x1x3'+x2x3'+x1'x2x3 For the LHS I can simplify it to: x2(x1'+x1x3') + x1'x3 + x1x2' x2x1' + x2x3' + x1'x3 + x1x2' x1'x3 + x2x3' + x1x2' For the...
  20. K

    Solving Boolean Algebra: a'b'c' + abc = 1?

    Hi guys, I'm new to boolean algebra, i couldn't get this through... a'b'c' + abc = 1 ? or i have to use (abc)' + abc = 1 to get 1 ? Thanks you.
  21. B

    Boolean Algebra Simplification: Understanding F= A'+B'+(A+B).B'.C [Tutorial]

    Digital Logic is blowing my head off! I can't understand a thing. I had a question in which I was supposed to simplify the function F= A'+B'+(A+B).B'.C apperently this simplify to 1. Here's what I managed to do: F= A'+B'+[(A.B')+(B.B')].C by Distribution F= A'+B'+(A.B').C by Tautology F=...
  22. M

    Simplifying equations using boolean algebra

    I need to simplify these boolean expressions. Homework Statement Problem A: a`b`d+a`bd`+bc`d`+bcd`+ab`c`d+b`cd Problem B: a`b`cd+a`bd+abcd+a`b`d+acd+abd+ab`c`d Problem C: a`b`c`d`+a`b`cd`+a`bd+ab`c`d`+bd+ab`d`Homework Equations abc means a AND b AND c a+b means a OR b a` means NOT a The...
  23. D

    Deriving full adder sum and carry outputs using boolean algebra

    Homework Statement Hi, I am trying to write the sum and output of a full adder in terms of XOR logical functions using boolean logic and Karnaugh maps. I've got the expressions from the Karnaugh maps fine but I can't seem to rearrange them into the expected form shown at the end of my...
  24. N

    Converting English to Boolean Algebra

    Homework Statement An irrigation system should open sprinkler's water valve when if the system is enabled and neither raining nor freezing temperatures are detected. Homework Equations S\rightarrow system enabled R'\rightarrow not raining F'\rightarrow freezing temperatures not detected...
  25. S

    Boolean Algebra Expansion and Equivalence with Truth Table

    [b]1. (A+B+C')(A'+B'+C)=A (XOR) B + (A XOR C)' + (B XOR C)' Expand right hand side first and then show tht its equal to left hand side by using truth table. I am unable to expand the right hand side i need help thnx
  26. L

    Difference between a boolean algebra and a complete lattice?

    What is the difference between a boolean algebra and a complete lattice? What are their similarities?
  27. H

    Proof of x + x'.y = x + y Using Boolean Algebra

    I was wondering how you can prove using Boolean algebra, x + x'.y = x + y I know we must manipulate either side of the equation but I'm just not sure how I can. Can anyone offer me some help.
  28. P

    Boolean Algebra- having trouble solving

    Homework Statement AB + (A' + B')C + AB Homework Equations I've simplified many problems, but this one is giving me trouble. I know the answer is AB + C, but for some mental reason, I can't seem to come to that answer. The Attempt at a Solution STEP 1: AB + A'C + B'C + AB...
  29. A

    Thanks.Boolean Algebra Reduction - Is There an Algorithm for It?

    Hi Everybody, I have a general question: Is there any Algorithm that can do Boolean Algebric Reduction ? Appreciate your help
  30. F

    Boolean Algebra optimization question

    Hi, This is a question about a boolean "law" type behavior I've noticed in my homework a couple of times. Basically i can't find a boolean algebra law that permits this optimization short of using a k-map. So I'm just wondering if theirs some way to optimize the one equation using...
  31. O

    Boolean Algebra: What Audio Engineering Majors at UMiami Need to Know

    what exactly is it? i have a friend that has to take it, he's an audio engineering major at UMiami. i forgot to ask him when i saw him last
  32. S

    Boolean Algebra: Simplifying x'yc+xy'c+xy to yc+xc+xy

    show that x'yc+xy'c+xy = yc + xc + xy The Attempt at a Solution - my first attempt at the solution was this: c(x'y+xy')+xc >>> c(y+x)+xc >>> yc+xc+xy I can't remember if that is valid or not. I seem to remember that A'B=B, but not sure. Can somebody please help me?
  33. M

    Simplifying (Boolean Algebra)

    Homework Statement Simplify these equations and functions 1. xyz’ + x’ (x + z) + x’yz + x’y’z’ 2. xy (z + z’) + x’y’z 3. wxyz + w’xy’z + wx (y’ + z) + w’x’z 4. F(A, B, C, D, E) = ∑(0, 1, 5, 6, 13, 15, 20, 21, 22) 5. F(w, x, y, z) = ∑(0, 1, 2, 3, 11, 13, 15) Homework Equations...
  34. quasar987

    Boolean Algebra Embeddings: Defining and Understanding the Role of Monomorphisms

    According to my notes, the definition of an embedding from a boolean algebra B in a boolean algebra B' is an injective map f:B-->B' such that for all x,y in B, f(sup{x,y}) = sup'{f(x),f(y)} and f(Cx)=C'(f(x)), where sup is the supremum in B and sup' is the complement in B', and where C is the...
  35. K

    Simplifying equations using boolean algebra

    [b]1. Simplify the following equations using boolean algebra [b]2. a) abc + ab'c'+ ab'c b) (abc)'+(a+c)'+b'c' [b]3. Please help me to solve the above equations
  36. K

    Simplify using Boolean algebra

    Simplify the following expressions using Boolean algebra. a. AB + A(CD + CD’) b. (BC’ + A’D) (AB’ + CD’)
  37. U

    EE - Building Boolean Algebra using CMOS Gates

    Homework Statement Sketch a CMOS logic circuit that realizes the function Y = AB + A'B'. This is called the equivalence or coincidence function. Homework Equations The Attempt at a Solution The top is connected to Vcc... Is this correct? I'm learning this on my own and...
  38. Amith2006

    Solving Boolean Algebra Expression: Y=ABCD-ABD

    Homework Statement 1) Simplify the following Boolean expression: Y = ABCD – ABD Homework Equations The Attempt at a Solution I simplified it in the following way: Y = ABCD – ABD. (C + 1) = - ABD But the answer given in my book is ABD.
  39. A

    A very quick 6 second question about boolean algebra.

    I was doing a problem, and the distributive problem hit my head, as I kind of blanked out on this one. Can B'D' + BD be simplified into a smaller equation? Because I'm thinking it's impossible to do so. (You may be able to apply Demorgan's theorem, but that doesn't really simplify the...
  40. A

    Boolean Algebra, Logic Diagram, K-Map, Nor gates HELP

    Boolean Algebra, Logic Diagram, K-Map, Nor gates... HELP! Homework Statement F(A,B,C,D) = Sigma(2,4,6,10,12) d(A,B,C,D) = Sigma(0,8,9,13) [Dont Care Functions] Implement the function using no more than 2 NOR gates. Homework Equations K-map The Attempt at a Solution First of all, I...
  41. K

    How to Simplify Boolean Algebra Expressions

    I just started Digital Systems coursework and it seems our professor felt like giving us a book that has no examples and progresses the class with no examples as well. So basically we are stuck trying to learn Boolean algebra reduction with no real guidence. Great. Anyways, I'm looking...
  42. S

    How can a logical approach help prove Boolean algebra properties?

    For my digital logic class, we are supposed to prove all of the two and three variable properties (commutative, associative, distributive...). I'm not really sure how to go about this, because although he didn't say, it seems intuitive to prove them in order, ie you can't use a property until...
  43. C

    Am I to the end of this Boolean Algebra Problem?

    Here's the problem: a'b'c'd' + a'b'cd' + a'b'cd + ab'c'd' + abc'd' I've gotten it down to: ac'd' + b'c'd' + a'b'c Having trouble coming up with a way to simplify it more...Is this as far as it can go? Any help appreciated, thanks...
  44. C

    Need Help with Boolean Algebra Equations - Proof and Explanation

    I would like you guys to check over my Boolean Algebra equations to see if I did them right. My teacher uses the same symbols as the computer science books Legend: http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL]=[/URL] XOR XOR =...
  45. M

    I'm stuck on boolean algebra, it has 4 variables

    Hello everyone, I'm designing a IC that will result in a 1 output if the number is > 9. So I wrote out the truth table and or'ed all the min terms and got the following: http://img282.imageshack.us/img282/482/lastscan45vd.jpg I'm stuck now! any help would be great!
  46. M

    Solve Boolean Algebra Problem: Output 1 for Numbers > 9 | IC Design Tips

    Boolean algebra problem! Hello everyone 'im designing a IC that will tell output a 1 if the number is > 9. so I wrote this boolean expression but I'm stuck on reducing it. B' = B compelemnted AB'CD' + AB'CD + ABC'D' + ABC'D + ABCD' + ABCD AB'(CD' + CD) + AB(C'D' + C'D + CD' + CD) any...
  47. M

    Problem with Simplifying Boolean Expression

    Hello everyone I'm having problems proving this: note: A' will stand for complemented; AB + BC'D' + A'BC + C'D = B + C'D B(A+C'D') B(A+C'+D') BA + BC' + BD' + A'BC + C'D BA + BD' + A'BC + C'(B+D) B(A + D' + A'C) + C'(B+D) now I'm stuck, i don't see how that is going to work out. Any...
  48. D

    Can Boolean Algebra Simplify Complex Logical Expressions?

    Boolean Algebra Hard! hey all, got stuck on some boolean algebra just wondering if you all can check my working out thanks :) basically the ` represent bar's and in the example no. 1 the p is barred, r is barred,and the overall function is barred hope you get the gist of things thanks 1...
  49. D

    Checking Boolean Algebra Solutions

    hey all, got stuck on some boolean algebra just wondering if you all can check my working out thanks :) basically the ` represent bar's and in the example no. 1 the p is barred, r is barred,and the overall function is barred hope you get the gist of things thanks 1. [(p` + qr)(pq+r`)]` my...
  50. C

    How to Solve a Boolean Equation with a Bar Above One Variable?

    I was just wondering if anyone could lend me some guidence in the boolean equation AB(A+B)= X when the last "B" in the formula has a bar above it.. thanks
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