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. sunrah

    Boolean Algebra (negation quickie)

    is this a valid statement \overline{A\overline{B}} = \overline{A}B
  2. C

    Boolean Algebra Simplification

    Homework Statement R=(A \cdot \overline{B})+(\overline{A} \cdot B) Using Boolean algebra, including the complement relation D\cdot \overline{D}=0, convert R to a form that uses one NAND, one AND and one OR gate (a total of three gates). Homework Equations de Morgan's theorem...
  3. T

    How Do You Simplify Complex Boolean Expressions?

    Homework Statement Ok so I'm having trouble simplifying a part of a boolean equation. It needs to be in the form Yt(...) Homework Equations note X' means Not X The part to be simplified is.. Y(NX'Z+NX'+NZ') The Attempt at a Solution Using the rules of simplification, particularly...
  4. T

    Boolean Algebra Simplification

    Homework Statement Hey there, I'm having trouble simplifying a boolean expression using XOR and XNOR functions. The final goal is to draw a logic circuit for the expression using NAND and XOR gates only.Homework Equations Assuming W' = Not WW' X' Y' Z' + W' X' Y Z + W' X Y' Z + W' X Y Z' + W...
  5. P

    Boolean Algebra in the Context of Mathematics

    I'm currently taking a course on Boolean algebra. It's officially named "Mathematical Logic", but I really don't see much resemblance between what I'm doing and other courses of mathematical logic that are taught in the US. It has only been a couple of weeks, but to let you have an inkling of...
  6. M

    Simplifying Boolean Algebra Expressions with K-maps

    Homework Statement Show that F=wy'z+x'y'z+w'xy+wx'y+wxy = y'z+yw+yx Homework Equations Boolean Algebra.. The Attempt at a Solution F=y'z(x+x') + yw(x+x') + w'xy => y'z+yw+w'xy => y'z + y(w+w'x) => y'z + y + yw <-----stuck here what do i do? Not sure if this step is...
  7. P

    How do you know if you can simplify boolean algebra more?

    How do you know if you can can or can not simplify a boolean algebra expression further?
  8. B

    Boolean algebra: Logical equivalence

    Homework Statement I've been trying to prove the equivalence between the two statements for quite a while now, any ideas? (A+C)(B+C') = BC + AC' Homework Equations The Attempt at a Solution I used the distributive property to simplify the LHS to AB + BC + AC' Unsure what...
  9. B

    Order of operations in boolean algebra

    I have a rather technical question in boolean algebra to which I was not able to find an answer. What is the order of operations in boolean algebra? Is it first logical NOT, logical AND, logical OR, and finally logical XOR? Correct me if I am wrong. BiP
  10. L

    Solving Boolean Algebra Problems: A+B = A+B

    Hello, Im trying to understand boolean algebra, I have to do the next problem using Boolean algebra and Identities for this one: So far this what I got: A + /A (A+B) = A + B A + /AA + /AB = A+B A + A + /AB = A + B A + A + B = A + B <-- I think A + A = A so .. A + B = A + B <-- I am not sure...
  11. K

    Boolean algebra simplification ?

    Homework Statement Homework Equations DeMorgan’s Theorems. The Attempt at a Solution I've had a go at it, not sure if I'm heading in the right direction though. Thanks for any help.
  12. twoski

    Boolean Algebra Equivalencies: Can These Expressions Be Simplified?

    Homework Statement Prove the following through boolean algebra x’y’z’ + xy’z + x’yz’ + xyz + xyz’ = x’z’ + yz’ + xz x’y’w + x’yw + x’yzw’ = x’w + x’yz xy’z + x’y’z + xyz The Attempt at a Solution Well i start by using the commutative law on the first one. x’y’z’ + xyz +...
  13. A

    How to simplify this boolean algebra (logic)

    This is related to logic circuits. How do I simplify this boolean algebra(logic) into its simplest expression? h = (abc + abd + acd + bcd)’ My first attempt would be to change it into product-of-sum. h = (abc)'(abd)'(acd)'(bcd)' and then what next?
  14. S

    Advice required on Boolean algebra question

    I have been asked to simplify this Boolean expression: ~A~B~C~D + ~A~B~CD + ~A~BCD + ~AB~C~D + ~AB~CD + ~ABC~D + ~ABCD + A~B~C~D + A~B~CD I have simplified it down to: ~AB + ~A~C + ~AD + ~C~B But I know that the simplification goes further to: ~AB + ~AD + ~C~B I've looked at...
  15. D

    Boolean Algebra simplification question

    Homework Statement Lets say I have 4 inputs x1 x0 y1 y0 if I have a sum-of-products say: x1x0'y1y0'+x0y1'y0 can I simplify it to x0y1'y0 by pulling out x0y1'y0 giving x0y1'y0(y0'+1) knowing that 1+y0 is 1 and 1*signalz is signalz. Am I right in thinking this, I really don't want to...
  16. D

    Boolean Algebra simplification question

    Homework Statement Lets say I have 4 inputs x1 x0 y1 y0 if I have a sum-of-products say: x1x0'y1y0'+x0y1'y0 can I simplify it to x0y1'y0 by pulling out x0y1'y0 giving x0y1'y0(y0'+1) knowing that 1+y0 is 1 and 1*signalz is signalz. Am I right in thinking this, I really don't want to...
  17. B

    Boolean Algebra: Proving A'''' = A Using Complement and Binary Addition

    Suppose that A is an n-digit binary string: a_{0}a_{1}a_{2}...a_{n} where a_{i} is a member of the set \{0,1\} Now define the following sequence of operations: 1) Complement each digit in A to get a binary string A' 2) A'' = A' + 1 (here + is taken to mean binary addition rather than the...
  18. B

    Boolean algebra: Simplifying logical expression

    I was trying a while ago to prove associativity of the XOR operator. That led me to the following problem: I am trying to prove \overline{(\overline{A}*B+\overline{B}*A)}*B + \overline{B}*(\overline{A}*B+\overline{B}*A) = B Obviously it can be solved by just creating a truth table...
  19. D

    Boolean Algebra simplification problem

    Homework Statement I not sure if this is the right place for this, so move me if you need. Ok let's say I have the function xyz+xy'z' - can I pull a x out giving x(yz+y'z') and say that yz=a giving x(a+a') which is x(1) or x. If this is true that means a=yz so a'=(yz)'. Can you distribute...
  20. B

    Boolean Algebra Simplification

    Homework Statement Simplify the following boolean expression: A'C + A'C'D + ABC + ABC'D I have a solution but when I checked against a boolean algebra simplifier online it gave me a different answer... is there a glaring mistake below ? Homework Equations None. The Attempt at...
  21. F

    Boolean Algebra Truth Table, Product and Sum Expressions

    Given f(x,y,z)=sum(1,2,3,4) draw truth table find product of sum exprsn find max terms find sum of product truth table x y z out 0 0 0 0 (x'+y'+z') 0 0 1 1 x'y'z 0 1 0 1 x'yz' 0 1 1 1 x'yz 1 0 0 1 xy'z' 1 0 1 0 (x+y'+z) 1 1 0 0 (x+y+z') 1 1 1 0 (x+y+z) sop=x'y'z...
  22. P

    Discreet Math question: Defining a boolean algebra on a set of function?

    I have a question on my homework that asks me to define a boolean algebra on set B where B = {f|f is a subset of integers squared and f is a function} I need to specify the operations, identity elements and show how to identify the complement elements. I have tried, but am really not...
  23. J

    Is This Boolean Expression and Logic Gate Diagram Correct?

    Homework Statement The question is shown in the picture attached. I have also attached my attempt of the circuit using logic gates. The Attempt at a Solution This is my answer: A’.B.C’.D’ + A’.B.C.D + A.B’.C’.D’ + A.B’.C’.D + A.B’.C.D’ + A.B.C.D’ = B.A’(C’.D’+C.D) + A.C.D’ +...
  24. C

    Boolean Algebra Proof Statement

    Homework Statement Prove the identity of the following Boolean equation using algebraic manipulation: AD' + A'B + C'D + B'C = (A' + B' + C' + D')(A + B + C + D) Homework Equations DeMorgan's Theorem: (A + B)' = A'B' The Attempt at a Solution I tried simplifying the equation to...
  25. X

    Basic Boolean Algebra Proof: Validity of the Cancellation Law?

    Homework Statement Prove that in a Boolean algebra the cancellation law does not hold; that is, show that, for every x, y, and z in a Boolean algebra, xy = xz does not imply y = z. Homework Equations The 6 postulates of a Boolean Algebra The Attempt at a Solution I am uncertain as...
  26. K

    Boolean algebra: I don't understand how the answer was achieved

    Homework Statement Factor to obtain a product of sums. (Simplify where possible.) BCD + C'D' + BC'D + CD Homework Equations Distributive: X(Y+Z)=XY+XZ and X+YZ=(X+Y)(X+Z) Simplification: XY+XY'=X and (X+Y)(X+Y')=X...
  27. H

    Boolean Algebra: Can't prove that ô.e+ô.ê+o.e=ô+e (the accent means negation).

    Consider a Boolean algebra (B,+,.,negation symbol). 0 is the zero element and 1 is the unit element. + has the lowest precedence and negation has the highest. Show that ô.e+ô.ê+o.e=ô+e for all o and e in B. This isn't a logic gate algebra because that'd be easy, so I can't use the values 0s...
  28. T

    Can someone confirm this? (About boolean algebra and reduced cost)

    I was asked to make the simplest possible circuit to solve this problem: for an entry of 0000 to 1001 (0 to 9), detect the numbers that are divisible by 2 or 3. I decided to write the truth table with an output of 1 for functions that are NOT divisible by either 2 or 3, since that seemed like...
  29. F

    Boolean Algebra Simplification Help

    Homework Statement A'B'C' + ABC' + A'BC + A'BC'. The ' denotes a bar over the previous letter. Homework Equations Simplification Rules The Attempt at a Solution C' (A'B' + AB) + A'B(C+C') C' (A'B' + AB) + A'B ... or like so: A'B'C' + ABC' + A'B Is it possible to simplify...
  30. T

    Solve Boolean Algebra Homework Problem

    Homework Statement I am suppose to use Boolean algebra to show that the following expressions are true. Homework Equations w'z' + w'xy + wx'z + wxyz = w'z' + xyz + wx'y'z + wyz The Attempt at a Solution I have tried to figure out how to get to that answer above but I am stuck...
  31. M

    Interpreting Non-English Administrative Acts with Boolean Algebra

    Hello all I don't write much, I mainly follow you silently. Great forum I have a doubt that to many of you will seem absolutely silly but it has been bugging me for days. I am trying to find the correct interpretation to a non-english administrative act and trying to use all possible...
  32. V

    Boolean Algebra Simplification

    Homework Statement Hey PF! I'm supposed to "Optimize the equation for minimal input-output delay with 3-input NAND gates of 1.8ns delay each." It'll become much clearer at my attempt at a solution, I hope. Homework Equations De Morgan's laws, K-maps, the sort... The Attempt at...
  33. Femme_physics

    Boolean Algebra VS Karnaugh map

    We were taught both methods to minimize gates. I frankly just want to pick one method all the time and become an expert in it, rather then try them both. So, according to your experience, which method do I better pick?
  34. H

    Using Boolean Algebra to Prove Equations

    Homework Statement There should be lines of some values to imply the "Not" form of them, however to make it easier, i'll just use the ¬ Symbol (a) Let x, y be elements of a Boolean algebra. Prove from the axioms that (x · y) + x = x. (b) Prove from the axioms of Boolean algebra that x...
  35. X

    Boolean Algebra - Simplest POS / SOP

    Hey all So i just need some clarification, though I hope I am posting in right section (this is not a HW Q) I was told to find the simplest Product of Sums(POS) and Sum of Products(SOP) 1st 2nd 3rd...
  36. M

    Boolean Algebra Trouble

    Homework Statement 1. Simplify the following problem using boolean algebra: (p+q'r')(p'q'+r) 2. Use algebra or karnaugh maps to simplify the following circuit: (z+xy')+yx+xzy'Homework Equations Boolean algebra rules and demorgan's lawThe Attempt at a Solution 1. (p+q'r')(p'q'+r)...
  37. M

    Boolean Algebra, Finding subscript for P and S

    Homework Statement Identify the equivalent binary numbers of the following P term and S terms( which are the subscripts of P's and S's) Ex: P?=AB'CD' Homework Equations No equations that I know of The Attempt at a Solution I create an table to find the subscripts for P and S...
  38. S

    Distributivity Theorem in Boolean Algebra

    Hello. I am wondering if the distributivity theorem works for factoring out more than a single term. Distributivity Theorem: BC + BD = B(C + D) But can I do this: [ tex ] ABC + ABD = AB(C + D) [ /tex ] Thanks.
  39. S

    Simple Boolean Algebra / Propositional Logic Question

    Homework Statement Produce the given truth table (given below as well as in a neater version in the attached Excel document) using the Boolean operators AND, OR, and/or NOT: A (Input 1) B (Input 2) O (Output) 1 1 0 1 0 0 0 1 1 0...
  40. romsofia

    Boolean Algebra: Harder Than Calculus/ODEs? Book Recs

    Is boolean algebra "hard" compared to calculus or ODE's? What's a rigid boolean algebra? And since I shouldn't start a new thread, can anyone recommend a good boolean algebra book? Thanks for your answers!
  41. T

    A boolean algebra manipulations (should be fun)

    Homework Statement Looking at the characteristic equation for a SR flip flop we have Q+ = QR¬ + S It's known (from the associated K-map of the SR/SC flippy floppy) that (Q+)¬ = Q¬S¬ + R K-maps are useful as they do the simplification for use, but I'm trying to use algebra to get from Q+ to...
  42. J

    Prove an Identity in Boolean Algebra: Help Needed!

    May seem easy, I can't do it though... I'm given an identity to prove:- (A + C).(notA + B) = A.B + notA.C I've started with LHS, multiplied out and ended up with an extra B.C. I think this has something to do with the distribution rule but I don't know how to work it through Any help...
  43. K

    Simplifying a Boolean Function: Is This the Correct Solution?

    I have the following function to be reduced/simplified. F(A,B,C,D) = BC + (A + C'D') where ' denotes the complement Here's my solution: = BC + (A + C'D')' = BC + (A + (C+D) = BC + (A + C + D) = BC + C + A + D = C(B + 1) + A + D = C*1 + A + D = C + A + D Is this correct?
  44. X

    How to apply De Morgan's theorem to a boolean equation?

    Homework Statement http://img525.imageshack.us/i/circuitg.png/ Get boolean equation and apply De Morgan. Homework Equations The Attempt at a Solution I'm not sure how to do this. Treat CLK like an variable? If so, is this correct? F = abC + AbC + ABC I do not know. I suppsoe you then...
  45. H

    Is B={0,1,R,F,X} a Boolean Algebra?

    Homework Statement is B={0,1,R,F,X) a boolean algebra? Use basic posulates/axioms to prove it. R=Rising F=Falling X=Dont Care Homework Equations Reference: Boolean Identities Table The Attempt at a Solution it is boolean algebra. you can create and, or , not tables with it...
  46. E

    Simplifying Boolean Expression with Boolean Algebra

    can anyone help me simplify this expression.. f(x,y,z)=(x+y+z')(x+y'+z')(x'+y'+z).. :confused:
  47. J

    A postulate of boolean algebra: x+(y.z) = (x+y).(x+z)

    x+(y.z) = (x+y).(x+z) I don't understand how one gets the RHS. Could you please it? There must be some reasoning involved. Please help me. Thanks.
  48. J

    Logic gates and boolean algebra

    Hi I'm not a science or physics student - quite a layman. Therefore, please keep your replies simple and explanatory so that your guidance and help can be fully appreciated. Thanks. What are these logic gates - AND, NOR, NAND, etc? Are these some physical circuits? In Boolean algebra...
  49. N

    How to Simplify Boolean Algebra: Can Someone Help Me with This Problem?

    hey there.. can someone help me with the problem given.. Problem.. simplify b ⊕ c' Homework Equations The Attempt at a Solution i don't know how to attempt this question.. can someone can give me an idea to start.. thanks in advance
  50. F

    Associative Law for Multiplication - Probability Theory and Boolean Algebra

    Hi, I'm having some trouble solving one of the problems from my homework assignment. Homework Statement Prove: P((AB)C) = P(A(BC)) Where A,B,C are either true or false. Homework Equations We can't do this by using a truth table, we can use the following equations: P(A + B) = P(A) + P(B) -...
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