Boolean Algebra differs from both general mathematical algebra and binary number systems. In Boolean Algebra, A+A =A and A.A = A, because the variable A has only logical value. It doesn’t have any numerical significance. In ordinary mathematical algebra, A+A = 2A and A.A = A2, because the variable A has some numerical value here. Also, in Binary Number System 1+1 = 10, and in general mathematical algebra 1+1 = 2 but in Boolean Algebra 1+1 = 1 itself. Unlike ordinary algebra and Binary Number System here is subtraction or division in Boolean Algebra. We only use AOI (AND, OR and NOT/INVERT) logic operations to perform calculations in Boolean Algebra.
布尔代数不同于一般的数学代数和二进制数系统。 在布尔代数中， A + A = A且AA = A ，因为变量A仅具有逻辑值。 它没有任何数值意义。 在普通的数学代数中， A + A = 2A且AA = A2 ，因为变量A在此处具有某些数值。 同样，在二进制数系统1 + 1 = 10中 ，通常在数学代数1 + 1 = 2中，但在布尔代数1 + 1 = 1中 。 与普通代数和二进制数系统不同，布尔代数是减法或除法。 我们仅使用AOI(AND，OR和NOT / INVERT)逻辑运算来执行布尔代数中的计算。
布尔代数中的公理 (Axioms in Boolean Algebra)
There are some set of logical expressions which we accept as true and upon which we can build a set of useful theorems. These sets of logical expressions are known as Axioms or postulates of Boolean Algebra. An axiom is nothing more than the definition of three basic logic operations (AND, OR and NOT). All axioms defined in boolean algebra are the results of an operation that is performed by a logical gate.
我们接受了一些逻辑表达式，这些逻辑表达式是正确的，并且可以在这些逻辑表达式上建立一组有用的定理。 这些逻辑表达式集被称为布尔代数的公理或假设 。 公理只不过是三个基本逻辑运算(AND，OR和NOT)的定义 。 布尔代数中定义的所有公理都是由逻辑门执行的运算的结果。
Axiom 1: 0.0 = 0 Axiom 6: 0+1 = 1 Axiom 2: 0.1 = 0 Axiom 7: 1+0 = 1 Axiom 3: 1.0 = 0 Axiom 8: 1+1 = 1 Axiom 4: 1.1 = 1 Axiom 9: 0 = 1 Axiom 5: 0+0 = 0 Axiom 10: 1 = 0
Based on these axioms we can conclude many laws of Boolean Algebra which are listed below,
根据这些公理，我们可以得出以下布尔代数的许多定律，
Commutative Laws
交换律
A+B = B+A, and A.B = B.A
Associative Laws
关联法
(A+B) + C = A+(B+C) (A.B). C = A. (B.C)
AND Laws
与法律
A.0 = 0 A.1 = A A.A = A A.A = 0
OR Laws
或法律
A+0 = A A+1 = 1 A+A = A A+A = 0
Complementation Laws
补充法
If A = 0 then A = 1 If A = 1 then A = 0 A̿ = A
Distributive Laws
分配法
A(B+C) = AB + AC A + BC = (A+B). (A+C)
Idempotence Law
幂等法
A.A = A, If A=1, then A.A = 1.1 =1 = A and if A=0, then A.A = 0.0 = 0 = A A+A = A, If A=1, then A+A = 1+1 =1 = A and if A=0, then A+A = 0+0 = 0 = A
Absorption Law
吸收定律
A + A.B = A A.(A+B) = A
De-Morgan’s Law
德摩根定律
A+B = A. B A.B = A + B
Consensus Theorem
共识定理
A) AB + A C + BC = AB + A C
Proof:
证明：
LHS = AB + A C + BC = AB + A C + BC (A+A) = AB + A C + ABC + ABC = AB (1+C) + A C (1+C) = AB + A C = RHS B) (A+B) (A + C) (B+C) = (A+B) (A + C)
Proof:
证明：
LHS = (A+B) (A + C) (B+C) = (AA + AC + BA + BC) (B+C) = (AC + BA + BC) (B+C) = ABC + ACC + BAB + BAC + BCB + BCC = ABC + AC + BA + BAC + BC + BC = ABC + AC + BA + BAC + BC = AC(B+1) + BA (1+C) + BC = AC + BA + BC …………. (Equation 1)RHS = (A+B) (A + C) = AA + AC + BA + BC = AC + BA + BC …………. (Equation 2)
Since, Equation 1 = Equation 2, Hence Consensus Theorem is verified.
由于等式1 =等式2 ，因此验证了共识定理 。
翻译自: https://www.includehelp.com/basics/axioms-and-laws-of-boolean-algebra.aspx
布尔代数定律
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