leftimage for poohprod.ru

Axioms - Laws of Boolean Algebra

Boolean algebra is the algebra of propositions. Propositions are denoted by letters, such as A, B, x or y, etc.

In the following axioms and theorems (laws of boolean algebra), the '+' or 'V' signs represent a logical OR (or conjunction), the '.' or '^' signs represent a logical AND (or disjunction), and '¬' or '~' represent a logical NOT ( or negation).

Every proposition has two possible values: 1 (or T) when the proposition is true and 0 (or F) when the proposition is false.

The negation of A is written as ¬A (or ~A) and read as 'not A'. If A is true then ¬A is false. Conversely, if A is false then ¬A is true.

Basic Laws of Boolean Algebra

Descript. OR form AND form Other way to express it:
Axiom x+0 = x x.1 = x A V F = A
A ^ T = A
Commutative x+y = y+x x.y = y.x A V B = B V A
A ^ B = B ^ A
Distributive x.(y+z) = (x.y)+(x.z) x+y.z = (x+y).(x+z) A ^ (B V C) = (A ^ B) V (A ^ C)
A V B ^ C = (A V B) ^ (A V C)
Axiom x+¬x = 1 x.¬x = 0 A V ¬A = T
A ^ ¬A = F
Theorem x+x = x x.x = x A V A = A
A ^ A = A
Theorem x+1 = 1 x.0 = 0 A V T = T
A ^ F = F
Theorem ¬¬x = x ¬(¬A) = A
Associativity x+(y+z) = (x+y)+z x.(y.z) = (x.y).z A V (B V C) = (A V B) V C
A ^ (B ^ C) = (A ^ B) ^ C
Absorption x+x.y = x x.(x+y) = x A V A ^ B = A
A ^ (A V B) = A
DeMorgan's laws x+y = ¬(¬x.¬y) x.y = ¬(¬x+¬y) A V B = ¬(¬A ^ ¬B)
A ^ B = ¬(¬A V ¬B)

Using logical gates, the commutative property for a logical AND is:

commutativity 1

The commutative property for a logical OR, is:

Using electronic gates, the distributive property is:

distributive property using gates

The De Morgan's laws can transform logical ORs into logical ANDs (negations are necessary) and can electronically be described this way:

De Morgan's law 1

De Morgan law 2

From 'Axioms' to home

From 'Axioms' to 'Boolean Algebra'


AND gate

OR gate

NOT gate

XOR gate

footer for boolean algebra axioms page

Related pages

convert numbers to asciibisection method algorithm matlabperiodic function definitioninterpolation lagrangehex to octal conversion methodcompound continuous interest calculatorbisection method examples with solutionspie chart matlabascii code for alphabets and numbersinterest compounded continuously formulainverse a matrix in matlabcalculating factorialzy smith chart colorhow to change from binary to decimalrc circuit practice problemsprintable ascii tablehow to find factorialsmatrices gauss jordankvl and kcl problems with solutionsdec to binary converterdemorgans rulesimpson's rule calculatormatlab interpolation functionsolving simultaneous nonlinear equationsmatrix is singular to working precisionascii char tablematlab lessonsintegrals matlablog function matlabmatlab visualize 3d matrixmatlab rgbhow to input matrix in matlabsolve equations matlabcircle in matlabstrings in matlabhanoi tower algorithm3-d plot matlabexponential matlabsolve two equations two unknowns matlabsolve definite integral calculatorperiodic functions exampleswriting function in matlabhex binary tabletrapezoidal rule code matlabparabolic interpolationonline smith chart plottermatlab matrix determinantsolving rc circuitsrc network calculatorrgb matlabcallback matlabmatlab plot tutorialexponential curve fitting matlab1089 trickxor exclusive orpower correction capacitorconversion of binary to octalsquare inches to square feet calculatorprogram for pascal trianglechi square online calculatorevaluate double integral calculatormatlab fit sine wavematlab program for unit step functionmatlab ideprime factorization exampleeuclidean algorithm gcfmatlab to excelformula for nominal interest ratesolved examples of nodal analysis3d plots in matlabrand matlabmaclaurin series for cosx3d plot matlabdecimal to binary conversion algorithm